Computational Chemistry

Transcription

1 Computational Chemistry Physical Chemistry Course Autumn 2015 Lecturers: Dos. Vesa Hänninen and Dr Garold Murdachaew Room B407

2 Contents of the course Lectures Principles of quantum chemistry Hartree Fock theory Electron correlation methods Density functional methods Semiempirical methods Graphical models Molecular dynamics Computer exercises Introduction to supercomputer environment (CSC) Introduction to computation chemistry programs Calculations of molecular properties Computer simulations of molecular structures, vibrations, chemical reactions,

3 What is computational chemistry? Relies on results of theoretical chemistry and computer science o o Theoretical chemistry is defined as mathematical description of chemistry. It is development of algorithms and computer programs to predict chemical properties. Computational chemistry is application of existing computer programs and methodologies to specific chemical problems. Practice of efficient computer calculations to obtain chemical and physical properties of molecules, liquids and solids + =

4 There are two different aspects to computational chemistry: 1. Computational studies can be carried out to find a starting point for a laboratory synthesis, or to assist in understanding experimental data, such as the position and source of spectroscopic peaks. 2. Computational studies can be used to predict the possibility of so far entirely unknown molecules or to explore reaction mechanisms that are not readily studied by experimental means. Thus, computational chemistry can assist the experimental chemist or it can challenge the experimental chemist to find entirely new chemical objects.

5 Importance of computational chemistry Computational methods are considered important in scientific community, see: The top 100 papers in science Top cited papers in physics of last 110 years The Most Cited Chemistry Papers Published, The most cited chemistry articles, 2005 Also, computational chemistry has featured in a number of Nobel Prize awards, most notably in 1998 and 2013.

6 Some applications Drug design Nanotechnological modelling and simulations Atmospheric science

7 Computational chemistry programs There are many self-sufficient software packages used by computational chemists. Some include many methods covering a wide range, while others concentrating on a very specific range or even a single method. Details of most of them can be found in: Biomolecular modelling programs: proteins, nucleic acid. Molecular mechanics programs. Quantum chemistry and solid state physics software supporting several methods. Molecular design software Semi-empirical programs. Valence bond programs.

8 Challenges and limitations Theory level: Is the precision high enough for the application? The computer programs: Efficiency and implementation Computer resources: Is there enough power for the task?

9 About CSC CSC IT Center for Science Ltd is administered by the Ministry of Education, Science and Culture. CSC maintains and develops the state-owned centralised IT infrastructure and uses it to provide nationwide IT services for research, libraries, archives, museums and culture as well as information, education and research management. Researchers can use it s large collection of scientific software and databases. CSC has offices in Espoo's Keilaniemi and in the Renforsin Ranta business park in Kajaani.

10 Methods of computational chemistry Ab initio that use rigorous quantum mechanics + accurate computationally expensive good results for small systems ~10 2 atoms Semi empirical that use approximate quantum mechanics relies on empirical or ab initio parameters + affordable and in some cases accurate limited to well defined systems with ~10 4 atoms Molecular mechanics that use classical mechanics relies on empirical force fields without accounting for electronic properties (no bond breaking or forming) + very affordable and used as a virtual experiment can handle extremely large systems ~10 9 atoms

11 Principles of quantum chemistry The state of the system is specified by a (normalized) wave function ψ For every measurable property (observable) of a system such as energy E for example, there exist a corresponding operator (H for E) Observables satisfy the eigenvalue equation. For example Hψ = Eψ. The expectation value of the observable, for example E, is given by or using Dirac s notation as E = ψ Hψ dτ E = ψ H ψ

12 Majority of computational chemistry revolves around finding a solution to the static Schrödinger equation Hψ = Eψ The list of closed-form analytic solutions is VERY short. The list of famous chemical problems includes the H atom, the harmonic oscillator, the rigid rotor, the Morse potential, and the ESR/NMR problem. Hydrogen atom atomic orbitals can be used as a basis for the molecular orbitals. Harmonic oscillator basis for the molecular vibrational motion Morse oscillator basis for the molecular stretching vibration Rigid rotor basis for molecular rotational motion

13 Variational method Yields approximate solution for the Schrödinger equation. Variational principle states that the expectation value of the Hamiltonian for trial wavefunction φ must be greater than or equal to the actual ground state energy φ H φ φ φ = E[φ] E 0 Example: Trial function expanded as a linear combination eigenfunctions of the hydrogen ground state ψ 0 and the first exited state ψ 1 φ = c0 ψ0 + c1 ψ1 where c 0 and c 1 are unknown coefficients, so called variational parameters. The hydrogenic energy corresponding to this trial function is E φ = c 0 2 ψ 0 H h ψ 0 + c 2 1 ψ 1 H h ψ 1 + c 0 c 1 ψ 0 H h ψ 1 + c 1 c 0 ψ 1 H h ψ 0 c 2 2 = c 0 2 E 0 + c 2 1 E c 1 c 2 2 E c 1

14 Any variations in the trial function φ = c 0 ψ 0 + c 1 ψ 1 which lower the energy expectation value are bringing the approximate energy closer to the exact value. The best solution can be obtained via optimization, i. e. searching the values of variational parameters c i which minimize the energy. E c i = 0 c 0 E For hydrogen atom, the optimal solution is obviously found when c 0 = 1 and c 1 = 0. c 1

15 In reality we don t know the eigenfunctions. Instead we use basis functions which are physically relevant for the problem in hand. φ = c 0 φ 0 + c 1 φ 1 The hydrogenic energy corresponding to this trial function is E φ = c 0 2 φ 0 H h φ 0 + c 1 2 φ 1 H h φ 1 + c 0 c 1 φ 0 H h φ 1 + c 1 c 0 φ 1 H h φ 0 c c 1 2 E c 0 = 2c 0 φ 0 H h φ 0 + 2c 1 φ 0 H h φ 1 c c 1 2 2c 0E c c 1 2 = 0 φ 0 H h φ 0 E c 0 + φ 0 H h φ 1 c 1 = 0 E c 1 = 0 φ 0 H h φ 1 c 0 + ( φ 1 H h φ 1 E)c 1 = 0

16 We have a system of linear equations. According to linear algebra the energy eigenvalues can be obtained by finding solutions of a characteristic equation φ 0 H h φ 0 E φ 0 H h φ 1 φ 0 H h φ 1 φ 1 H h φ 1 E = H 0,0 E H 0,1 H 0,1 H 1,1 E = 0 The energies can be obtained by diagonalizing the Hamiltonian matrix H = φ 0 H h φ 0 φ 0 H h φ 1 φ 0 H h φ 1 φ 1 H h φ 1 = H 0,0 H 0,1 H 0,1 H 1,1 diag E E 1 In general when we have N basis functions determinant of a characteristic polynomial becomes H 0,0 E H 0,N H 0,N H N,N E = 0 The secular determinant for N basis functions gives an N-th order polynomial in which is solved for N different roots, each of which approximates a different eigenvalue.

17 Example: For a helium atom we can choose the trial function as follows: φ r 1, r 2 = C 1 + pr 12 + q r 1 r 2 2 exp α r 1 + r 2 where C is normalization constant and p, q, and α are variational parameters. After optimization: p = 0.30, q = 0.13, and α = E = a.u. (Three parameters) E = a.u. (1024 parameters) E = a.u. (Experimental value) Using one parameter trial function φ = C exp α r 1 + r 2 energy is the minimum E = a.u.

18 Born-Oppenheimer approximation Hamilton operator includes the kinetic and potential energy parts of the electrons and nuclei H = T + V The kinetic energy operator for electrons and nuclei can be written as n i 2 N j 2 T = ħ 2 i m e ħ 2 j m n where i 2 = 2 x i y i z i 2, n and N are numbers, and m e and m n are masses of electrons and nuclei, respectively. The potential energy operator includes the electron-electron, nuclei-nuclei, and electron-nuclei parts n V = 1 2 i n i i e 2 4πε 0 r i r i N j N j j Z j Z j e 2 4πε 0 R j R j n i N j Z j e 2 4πε 0 r i R j Where r i and R j are positions of electrons and nuclei and Z j is the atomic number of nuclei j. In practice, it is impossible to solve the Scrödinger equation for the total wavefunction Ψ(r i, R j ) exactly.

19 Practical solution: Let s approximate the wavefunction in a form, where it is factorized in electronic motion and nuclear motion parts ψ(r i, R j ) ψ el (r i ; R j )ψ n (R j ) where function ψ(r i ; R j ) describes electronic motion (depending parametrically on the positions of nuclei) and function ψ(r j ) describes the nuclear motions (vibrations and rotations). With this assumptions, the problem can be reformulated to two separate Scrödinger equations: H el ψ el r i ; R j H n ψ n R j = V(R j )ψ el r i ; R j = E n ψ n R j The former equation is for the electronic problem, considering the nuclei to be fixed. The eigenvalue V(R j ) can be called as interatomic potential, which is then used as a potential energy for the latter equation for the nuclear motion problem. This procedure, the so called Born-Oppenheimer approximation, is justified because electron is lighter than the proton by the factor 2000, the electron quickly rearranges in response to the slower motion of the nuclei.

20 Example: For the H 2 + -ion the total energy operator can be written as Using the Born-Oppenheimer approximation, we can write the electronic Hamiltonian (further simplified by using atomic units a.u. or Hartree) as H el = r 1 1 r 2

21 The ground state trial wave function is ψ = c ψ 1s1 + ψ 1s2 where the 1s functions are the type ψ 1si = 1 π 1 a e r i a 0 Where constant a 0 is the Bohr radius. The normalization is c = S 12 Where the S 12 is the overlap integral between ψ 1s1 and ψ 1s2 functions

22 Note that because R = r 1 r 2, the overlap integral and thus the ground state electronic energy depends parametrically on the distance between the two nuclei R V(R) R/bohr

23 The BO approximation is justified when the energy gap between ground and excited electronic states is larger than the energy scale of the nuclear motion. The BO approximation breaks down when for example in metals, some semiconductors and graphene the band gab is zero leading to coupling between electronic motion and lattice vibrations (electron-phonon interaction) electronic transitions becomes allowed by vibronic coupling (Herzberg-Teller effect) ground state degeneracies are removed by lowering the symmetry in non-linear molecules (Jahn-Teller effect) interaction of electronic and vibrational angular momenta in linear molecules (Renner-Teller effect) The Jahn Teller effect is responsible for the tetragonal distortion of the hexaaquacopper(ii) complex ion, [Cu(OH 2 ) 6 ] 2+, which might otherwise possess octahedral geometry. The two axial Cu O distances are 238 pm, whereas the four equatorial Cu O distances are ~195 pm.

24 In the vicinity of conical intersections, the Born Oppenheimer approximation breaks down, allowing non-adiabatic processes to take place. The location and characterization of conical intersections are therefore essential to the understanding to a wide range of reactions, such as photo-reactions, explosion and combustion reactions, etc.

25 Electron spin Electron spin is introduced in quantum equations by writing electron wavefunction as a product of spatial wavefunction ψ(r) and spin wavefunction. For example H atom 1s eigenfunctions take the form ψ r, σ = 1 π 1 a e r a 0α ψ r, σ = 1 π 1 a e r a 0β where σ is non-spatial spin variable. These two eigenfunction have the same energy because the total energy operator does not depend on spin. Integration of spin wavefunction over spin variable is defined formally as α α dσ = 1, β β dσ = 1, α βdσ = 0, and β αdσ = 0

26 Wave functions describing many-electron systems must change sign (be antisymmetric) under the exchange of any two electrons We describe n-electron wavefunction using the notation ψ 1,2,, n = ψ r 1 σ 1, r 2 σ 2,, r n σ n. The position variables are supressed in favor of keeping track of the electrons. The antisymmetric two-electron wave function must satisfy: ψ 1,2 = ψ 2,1 For example if we write the antisymmetric wavefunction for ground-state of Helium as ψ 1,2 = ψ 1s 1 α 1 ψ 1s 2 β 2 ψ 1s 2 α 2 ψ 1s 1 β 1 Clearly we see that if the coordinates of electrons have the same values, the wavefunction vanishes ψ 1,2 = ψ 1s 1 α 1 ψ 1s 1 β 1 ψ 1s 1 α 1 ψ 1s 1 β 1 = 0 This obeys the so called Pauli exclusion principle: Two identical fermions (particles with half-integer spin) cannot occupy the same quantum state simultaneously.

27 For n-electron system the antisymmetric wave function can be written as Slater determinants and have the form ψ 1,2,, n = 1 n! ψ 1 1 α 1 ψ 1 1 β 1 ψ m 1 β 1 ψ 1 2 α 2 ψ 1 2 β 2 ψ m 2 β 2 ψ 1 n α n ψ 1 n β n ψ m n β n where m = n 2 if n is even and m = n if n is odd. It has been shown that the Pauli exclusion principle is responsible for the fact that ordinary bulk matter is stable and occupies volume. The electrons of each atom cannot all fall into the lowest-energy orbital and must occupy successively larger shells. Atoms therefore occupy a volume and cannot be squeezed too closely together. The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or Coulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place at the same time.

28 Potential energy surfaces The potential energy surface (PES) is a central concept in computational chemistry. A PES is a relationship between energy of a molecular system and its geometry. The BO-approximation makes the concept of molecular geometry meaningful, makes possible the concept of PES, and simplifies the application of the Scrödinger equation to molecules. Since the atoms are motionless while we hold the molecule at any given geometry, its energy is not kinetic and it is by default potential ( depending on position ). The geometry of the molecule is defined using appropriate coordinate system (cartesian coordinates, internal coordinates, Jacobi coordinates, etc.). The so called reaction coordinate, which is important in describing the energy profile of a chemical reaction, is simply some combination of bond distances and angles.

29 Among the main tasks of computational chemistry are to determine the structure and energy of molecule and of the transition states (TS) involved in chemical reactions. The positions of the energy minima along the reaction coordinate give the equilibrium structures of the reactants and products. Similarly, the position of the energy maximum defines the transition state. Reactants, Products, and transition states are all stationary points on the potential energy surface. This means that for system with N atoms all partial derivatives of the energy respect to each of the 3N 6 independent geometrical coordinates (R i ) are zero: V R i = 0 i = 1,2,, 3N 6

30 In the one-dimensional case, or along the reaction coordinate, reactants and products are located in the energy minima and are characterized by a positive second energy derivative d 2 V dr 2 > 0 The transition state is characterized by a negative second energy derivative d 2 V dr 2 < 0 In the many-dimensional case, each independent geometrical coordinate, R i, gives rise to 3N 6 second derivatives: 2 V, 2 V,, 2 V. Thus, it is not possible to say whether any given R i R 1 R i R 2 R i R 3N 6 coordinate corresponds to a stationary point. To see the correspondence, a new set of coordinates ξ i, referred as normal coordinates, is used. They have the property that their cross terms or non-diagonal terms in second energy derivative matrix vanish: 2 V = 0, etc. For the energy minima: ξ 1 ξ 2 2 V ξ i 2 > 0 i = 1,2,, 3N 6

31 Stationary points for which all but one of the second derivatives positive are so-called saddle points and may correspond to transition states. If they do, the normal coordinate for which the second derivative is negative is referred to as the reaction coordinate ξ r : 2 V ξ r 2 < 0

32 Geometry optimization is the process of starting with an input structure quess and finding a stationary point on the PES. It s usually checked whether the stationary point is a minimum or a transition state by calculating its vibrational frequencies. In transition state one of the vibrations will possess negative (or imaginary) harmonic frequency. Electronic energy ZPVE corrected potential energy or enthalpy difference H The free energy of the system, which drives the chemical reaction, is strictly speaking not the potential energy calculated from the electronic Scrödinger equation. As we would expect from Heisenberg s uncertainty principle molecules vibrate incessantly even in the 0 K. Moreover, in the higher temperatures the entropy also contributes. Thus, when calculating meaningful energy differences and thermodynamic properties the vibrational energy levels are required. Specifically, the zero point vibrational energy (ZPVE) is needed to obtain accurate energy differences for example in the rate constant calculations.

33 The vibrations of molecules The harmonic vibrational frequency for a diatomic molecule A-B is ν = 1 2π k μ where k is the force constant, which is the second derivative of the potential energy with respect to the bond length R and μ is the reduced mass k = d2 V R dr 2 μ = m Am B m A + m B

34 Polyatomic systems are treated in similar manner. Here, the force constants are second energy derivatives respect to the normal coordinates. A nonlinear molecule with N atoms has 3N-6 (linear molecule has 3N-5) vibrational degrees of freedom (or vibrational modes). In the harmonic approximation, the potential energy is V q 1, q 2,, q N N = 1 2 i=1 N j=1 2 V q i q j q i q j Where q i are displacements in some appropriate coordinate system (for example internal coordinate displacements). We can find a new set of coordinates (Wilson s GF method) that simplify the above equation to the form: V ξ 1, ξ 2,, ξ N N = 1 2 i=1 2 V ξ i 2 ξ i 2 Where ξ i are the normal coordinates of the molecule. All normal modes are independent in the harmonic approximation.

35 Vibrations of a methylene group (-CH 2 -) in a molecule for illustration Symmetrical stretching Asymmetrical stretching Bending Rocking Wagging Twisting

36 Hartree-Fock theory In computational chemistry, the Hartree-Fock method has central importance If we have the HF solution, the accuracy can systematically be improved by applying various techniques It is based on variational approach HF is an approximate method, close in spirit to the mean-field approach widely used in solid state and statistical physics The Hartree method 1928 D.R. Hartree ( ) Cambridge, UK The Hartree-Fock method 1930 V.A. Fock ( ) Leningrad, Russia

37 Five major simplifications: 1. The Born-Oppenheimer approximation is inherently assumed. 2. Relativistic effects are completely neglected. 3. Each energy eigenfunction is assumed to be describable by a single Slater determinant. 4. The mean field approximation is implied. 5. The variational solution is assumed to be a linear combination of a finite number of basis functions.

38 Employing BO-approximation the (non-relativistic) electronic Schrödinger equation can be written as n n 2 2 A n Z A R An 2 r nm n m n ψ = Eψ When wavefunction ψ is expressed as a single slater determinant (we approximate the real wavefunction using products of single electron wavefunctions) of doubly occupied spatial orbitals we can write the energy as E = 2 i ε i 0 + i j 2J ij K ij where the first zeroth-order energy term ε 0 i is the energy without electron-electron repulsion, J ij is the Coulomb integral and K ij is the exchange integral J ij = ψ i 1 ψ j 2 1 r 12 ψ j 2 ψ i 1 dτ 1 dτ 2 K ij = ψ i 1 ψ j 2 1 r 12 ψ i 2 ψ j 1 dτ 1 dτ 2

39 We now wish to find the orbitals which lead to a minimum value of the energy. The treatment is simplified if we define coulomb and exchange operators as follows J j ψ i 1 = ψ j 2 1 r 12 ψ j 2 dτ 2 ψ i 1 K j ψ i 1 = ψ j 2 1 r 12 ψ i 2 dτ 2 ψ j 1 Note that the exchange operator K j exchanges electron 1 and 2 between the two orbitals ψ i and ψ j. The coulomb and exchange integrals can be written as J ij = ψ i 1 J j ψ i 1 dτ 1 K ij = ψ i 1 K j ψ i 1 dτ 1

40 It is seen that the coulomb operator is potential energy which would arise from interaction between electron 1 and an electron 2 with electron distribution ψ j 2 2. Such operators represent effective potentials for an electron moving in the repulsive field of other electrons. The exchange operator has no classical analog, since it arises from the nonclassical antisymmetry principle. The Hartree-Fock equation for some space orbital ψ i occupied by electron 1 is h j 2J j K j ψ i 1 = F 1 ψ i 1 = ε i ψ i 1 Where h 1 0 h 1 0 ψ i 1 = ε i 0 ψ i 1 is the hydrogen-like Hamiltonian operator for electron 1 in the field of bare nucleus (or nuclei), F 1 is Fock operator and ε i is the orbital energy. These equations show very clearly that in order to solve the one-electron orbital ψ i 1, it is necessary to know wavefunctions ψ j in order to set up the operators J j and K j. The orbitals ψ i 1 only account for the presence of other electrons in an average manner (mean-field theory). The Hartree Fock method is also called the self-consistent field method (SCF), meaning that the final field as computed from the charge distribution is required to be "self-consistent" with the assumed initial field.

41 When the Hartree-Fock equations are solved by numerical integration methods, the procedure in unwieldy. Even worse, the method is incapable of being extended to molecules (molecular orbital theory). The key development (presented by Roothan) was to expand the orbitals ψ i as a linear combination of a set of oneelectron basis functions. Introducing a basis set transforms the Hartree-Fock equations into the Roothaan equations. Denoting the atomic orbital basis functions as φ k, we have the expansion ψ i = c ki φ k k This leads to F c ki φ k = ε i c ki φ k k k Left multiplying by φ l and integrating yields a matrix equation c ki φ l Fφ k dτ = c ki F kl = ε i c ki φ l φ k dτ = ε i c ki S kl k k k k

42 The Roothaan equations form a set of linear equations in the unknowns c ki. For a nontrivial solution, we must have det F kl ε i S kl = 0 It is a secular equation which roots give the orbital energies ε i. The matrix equation can be written shortly as Fc = Scε Where F is the Fock matrix, S is the overlap matrix and ε is a diagonal matrix of the orbital energies ε i. The Roothan equations must be solved by an iterative process: One starts with guesses for the orbital expressions as linear combinations of the basis functions ψ j. This initial set of orbitals is used to calculate the Fock operator F. The matrix elements are calculated, and the secular equation is solved to give an initial set of ε i s; these ε i s are used to solve matrix equation for an improved set of coeffifients c ki, giving an improved orbitals ψ i, which are the used to calculate an improved F, and so on. One keeps going until no improvement in orbital coefficients and energies occurs from one cycle to the next.

43 Example: Helium atom ground state SCF calculation Basis set of two 1s STOs: φ k = 1 π ζ k a e ζ kr a 0, k = 1,2, ζ 1 = 1.45, ζ 2 = 2.91 By trial and error, these have been found to be the optimum ζ s to use for this basis set. The starting ground state wavefunction: ψ j = c 1 φ 1 + c 2 φ 2 with initial guess: c 1 = and c 2 = The Roothan matrix elements F kl = φ k F φ l = ε kl φ k J j φ l φ k K j φ l J j φ l 1 = ψ j 2 1 r 12 ψ j 2 dτ 2 φ l 1 K j φ l 1 = ψ j 2 1 r 12 φ l 2 dτ 2 ψ j 1 Note that the coulomb and exchange operators depend on our initial guess.

44 The initial estimate of secular equation det F kl ε i S kl = 0 is ε i ε i ε i ε i = 0 ε 1 = ε 2 = Substitution of the ε 1 into the Roothan equation gives improved orbital ψ i = 0.836φ φ 2 After few cycles the converged roots from secular equation are ε 1 = and ε 2 = and the final orbital is ψ i = 0.842φ φ 2. To obtain the ground state energy of the system one needs to add to the orbital energy (lower energy root ε 1 ) additional core electron energy term and in the case of molecules, a nuclear-nuclear repulsion potential energy. So the SCF energy for helium becomes E HF = = hartrees The limiting HF energy found with five basis functions is only hartrees lower in energy. The comparison to variational and experimental energies shows an approximate error of hartrees (110 kj/mol), which arises due to lack of electron correlation.

45 Input: Atom coordinates, the atomic number of the atoms, electron basis set Calculate Coulombic and exchange contributions Form Fock matrix using first guess orbitals Diagonalize Fock matrix and obtain improved energies and orbitals Repeat cycle until energies and orbitals remain unchanged Output: Energy, forces, electronic structure, etc.

46 Recall that the electron WF is characterized by spatial and spin variables If the number of electrons is even and orbitals are doubly occupied, we have a closed shell (Fig. a) If the number of electrons is odd, we have an open-shell system (Fig. b) In general, if the numbers of electrons with spins up and down are different, we have an open-shell system

47 Open-shell systems can be dealt with by one of two HartreeFock methods: Restricted open-shell Hartree Fock (ROHF) Unrestricted Hartree Fock (UHF) UHF theory is the most commonly used when the number of electrons of each spin are not equal. While RHF theory uses a single molecular orbital twice, UHF theory uses different molecular orbitals for the α and β electrons. The result is a pair of coupled Roothaan equations F α c α = Sc α ε α F β c β = Sc β ε β The pair of equations are coupled because each orbital has to be optimised in the average field of all other electrons. This yields sets of molecular orbitals and orbital energies for both the α and β spin electrons.

48 UHF method has one drawback. A single Slater determinant of different orbitals for different spins is not a satisfactory eigenfunction of the total spin operator -S 2 The electronic state can be contaminated by excited states. For example, the doublet state (one more α spin electron than β spin) may have too large total spin eigenvalue. If S 2 = 0.8 or less 1 (exact value is = 0.75), it is probably satisfactory. If it is 1.0 or so, it is certainly not 2 2 satisfactory and the calculation should be rejected and a different approach taken. Given this drawback, why is the UHF used in preference to the ROHF? UHF is simpler to code it is easier to develop post-hartree Fock methods it is unique, unlike ROHF where different Fock operators can give the same final wave function.

49 Water dissociation Large systematic error in HF electronic energy The equilibrium structure and the shape of the potential energy surface near the equilibrium position is fairly well described The RHF method fails when atoms gain ionic character The UHF can be used to calculate relative energy changes when molecule dissociates

50 Helium atom Hamiltonian H = r 1 r 2 r 12 True wavefunction posesses a coulomb hole at r 1 = r 2 and α = 0. The HF wavefunction (expressed as a single Slater determinant) has no special behaviour near coalescence, i.e. No electron correlation In reality, electrons tend to avoid each other. Unphysically large propabilities to finding two electrons near eachother results in overestimated potential energies in Hartree-Fock calculations

51 Lack of electron correlation in HF leads to an error of 1 ev (100 kj/mol) per (valence)electron pair. In some cases the errors cancel themselves out Isodesmic reactions wherein the number and type of bonds on each side of the reaction remains unchanged can be calculated within few kcal/mol Example: Reaction of ethanol with methane ethanol + methane methanol + ethane The heat of formation of ethanol can be estimated now by simply calculating the reaction energy, with quantum mechanical methods and by using the computed reaction energy together with the known heats of formation. Using cheaply calculated HF/STO-3G energies for all four species, a reaction energy of kj/mol is predicted. Together with the know heats of formation, a value of -220 kj/mol is predicted for ethanol. This has to be compared to the experimental value of kj/mol.

52 Hartree-Fock and experimental equilibrium bond lengths R e (in pm) Molecule Bond HF Exp. H 2 R HH HF R FH H 2 O R OH O 3 R OO CO 2 R CO C 2 H 4 R CC CH 4 R CH Hartree-Fock calculations systematically underestimate equilibrium bond lengths The HF results are satisfactory

53 Hartree-Fock and experimental electronic atomization energies (kj/mol) Molecule HF Exp. F H HF H 2 O O CO C 2 H CH Hartree-Fock calculations systematically underestimate atomization energies Hartree-Fock method fails to describe correctly the electronic structures of some molecules such as diatomic fluorine and ozone.

54 Hartree-Fock and experimental electronic reaction enthalpies (kj/mol) Reaction HF Exp. CO + H 2 CH 2 O H 2 O + F 2 HOF + HF N 2 +3H 2 2NH C 2 H 2 +H 2 C 2 H CO 2 +4H 2 CH 4 + 2H 2 O CH 2 C 2 H O 3 +3H 2 3H 2 O Hartree-Fock method fails when reaction is far from isodesmic Some results are suprisingly good

55 Hartree-Fock summary Electronic state described by single Slater determinant Electronic ground state Each electron move in mean field created by other electrons Electron correlation is largely neglegted which can lead to large deviations from experimental results. Based on variational approach The best possible solution is at the Hartree-Fock limit: Energy as the basis set approaches completeness. HF method provides the starting point for the methods that take the electron correlation into account (the post-hf methods). The model can be systematically improved by applying corrections.

56 Electron correlation In HF the electronic wave function is approximated by a single Slater determinant Not flexible enough to account for electron correlation therefore the Hartree-Fock limit is always above the exact energy Some electron correlation is already found in the electron exchange term Better description for the wavefunction needed for more accurate results

57 Some common nomenclature found in literature: Fermi correlation arises from the Pauli antisymmetry of the wave function and some of it is taken into account already at the single-determinant level. For example the wavefunction for ground-state Helium is ψ 1,2 = ψ 1s 1 α 1 ψ 1s 2 β 2 ψ 1s 2 α 2 ψ 1s 1 β 1 where 1 and 2 in parentheses denote the coordinates of electrons 1 and 2, respectively. Clearly we see that if the coordinates of electrons have the same values, the wavefunction vanishes ψ 1,2 = ψ 1s 1 α 1 ψ 1s 1 β 1 ψ 1s 1 α 1 ψ 1s 1 β 1 = 0 The wavefunction vanishes at the point where the two electron coincide. Around each electron there will be a hole in which there is less electron with a same spin: the Fermi hole. This exchange force is comparatively localized.

58 Static correlation refers to situations in which multiple determinants are required to cover the coarse electronic structure. Electronic state is described by a combination of (qualitatively) different configurations which have comparative weights. Static correlations deals with only few, but very important determinants. Multi-configurational self-consistent field (MCSCF) is used to handle such correlations. Bond dissociation Excited states Near-degeneracy of electronic configurations (for example a singlet diradical CH 2 ). Dynamical correlation arises from Coulombic repulsion. To account for this, many determinants with small weights each are required. Dynamical correlation is needed to get the energetics of a system right, but not for the coarse electronic structure. Quite predictable, the major contribution is around 1 ev for each closed shell pair. Well accounted for by DFT functionals, perturbation theory, configuration interaction, and coupled cluster methods. Coulomb hole: The probability of finding two electrons at the same point in space is 0 as the repulsion becomes infinite. For the wavefunction approximated by Hartree-Fock method this requirement is not fulfilled. There is no explicit separation between dynamical and static correlations.

59 Hartree-Fock wave function of He atom Exact wave function of He atom nucleus electron 2 nucleus electron 2 the wave function of electron 1 while keeping the another fixed at x=0.5 Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

60 Correlation hole Difference between the exact and HF wave functions for the He atom nucleus electron 2

61 Configuration interaction method Configuration interaction (CI) has the following characteristics: A post-hartreefock linear variational method. Solves the nonrelativistic Schrödinger equation within the BO approximation for a multielectron system. Describes the linear combination of Slater determinants used for the wave function. o Orbital occupation (for instance, 1s 2 2s 2 1p 1...) interaction means the mixing of different electronic configurations (states). In contrast to the HartreeFock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of configuration state functions (CSFs) built from spin orbitals: ψ = i C i ψ i = C 0 ψ 0 + C 1 ψ 1 + C 2 ψ 2 +

62 Coefficients from the wavefunction expansion are determined by a variational optimization respect to the electronic energy HC = E CI C where H is the Hamiltonian matrix with matrix elements H ij = ψ i H ψ j The construction of the CI wavefunction may be carried out by diagonalization of the Hamiltonian matrix, but in reality iterative techniques are used to extract eigenvalues and eigenfunctions (Newton s method). The first term in the CI-expansion is normally the Hartree Fock determinant ψ = C 0 ψ HF + C 1 ψ 1 + C 2 ψ 2 +

63 The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree Fock determinant If only one spin orbital differs, we describe this as a single excitation determinant If two spin orbitals differ it is a double excitation determinant and so on The eigenvalues are the energies of the ground and some electronically excited states. By this it is possible to calculate energy differences (excitation energies) with CI-methods.

64 The expansion to the full set of Slater determinants (SD) or CSFs by distributing all electrons among all orbitals is called full CI (FCI) expansion. FCI exactly solves the electronic Schrödinger equation within the space spanned by the one-particle basis set. In FCI, the number SDs increase very rapidly with the number of electrons and number of orbitals. For example, when distributing 10 electrons to 10 orbitals the number of SDs is This illustrates the intractability of the FCI for any but the smallest electronic systems. Practical solution: Truncation of the CI-expansion. Truncating the CI-space is important to save computational time. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. These methods, CID and CISD, are in many standard programs. CI-expansion truncation is handled differently between static or dynamical correlation. In the treatment of static correlation in addition to the dominant configurations, near degenerate configurations are chosen (referred as reference configurations). Dynamical correlation is subsequently treated by generating excitations from reference space.

65 Excitation energies of truncated CI-methods are generally too high because the excited states are not that well correlated as the ground state is. The Davidson correction can be used to estimate a correction to the CISD energy to account for higher excitations. It allows one to estimate the value of the full configuration interaction energy from a limited configuration interaction expansion result, although more precisely it estimates the energy of configuration interaction up to quadruple excitations (CISDTQ) from the energy of configuration interaction up to double excitations (CISD). It uses the formula: δe Q = 1 C 0 2 E CISD E HF where C 0 is the coefficient of the Hartree Fock wavefunction in the CISD-expansion

66 CI-methods are not size-consistent and size-extensive Size-inconsistency means that the energy of two infinitely separated particles is not double the energy of the single particle. This property is of particular importance to obtain correctly behaving dissociation curves. Size-extensivity, on the other hand, refers to the correct (linear) scaling of a method with the number of electrons. The Davidson correction can be used. Quadratic configuration interaction (QCI) is an extension CI that corrects for size-consistency errors in the all singles and double excitation CI methods (CISD). This method is linked to coupled cluster (CC) theory. o Accounts for important four-electron correlation effects by including quadruple excitations

67 Example: CI calculation for Helium atom Lets us begin with two-configuration wavefunction expressed as a linear combination of hydrogenic wavefunctions having the form ψ 1,2 = c 1 ψ 1 + c 2 ψ 2 where ψ 1 arises from the configuration 1s 2 and ψ 2 arises from the configuration 1s2s. Specifically, the two wavefunctions are ψ 1 = 1 2 1s 1 α 1 1s 1 β 1 1s 2 α 2 1s 2 β 2 ψ 2 = 1 2 1s 1 α 1 1s 1 β 1 2s 2 α 2 2s 2 β s 1 α 1 2s 1 β 1 1s 2 α 2 1s 2 β 2

68 Since both ψ 1 and ψ 2 describe singlet states, there will be no vanishing matrix elements of H. If we represent these matrix elements by H ij = ψ i H ψ j (i, j = 1 or 2), the secular determinant to be solved is H 11 E H 12 H 12 H 22 E = 0 The diagonal matrix elements H 11 and H 22 are just the energies of single configurational calculations for the ground and exited states. Keeping in mind that the spin portions integrate out separately to unity α α = β β = 1, α β = 0 we obtain for the H 11 H 11 = 1s 1 1s 2 H 1s 1 1s 2 = 2ε 1 + J 11 where we represent the 1s orbital by subscript 1. The energy ε has the same form as the hydrogen atom energy ε n = Z2 2n 2 = 2 n 2

69 Similarly we can obtain the expressions for the matrix elements H 22 and H 12 but this derivation is omitted here for the sake of simplicity and only the final results are shown. After the appropriate integrations, the matrix elements become H 11 = 2.75 H 22 = H 12 = The roots of the quadratic formula (produced by secular determinant) are E 1 = and E 2 = The lower root represents the ground state whose experimental energy is Note that this improves the single configurational result The higher root represents the lowest excited state (experimental energy = 2.15).

70 Compared to CISD-method, the simpler and less computationally expensive MP2-method gives superior results when size of the system increases (MP2 is size extensive). For water monomer, MP2 recovers 94% of correlation energy which remains similar with increasing system (cc-pvtz basis). For stretched water monomer (bond length doubled) CISD recovers only 80.2% of the correlation energy. o for a large variety of systems it recovers 80-90% of the available correlation energy o With the Davidson correction added, the error is reduced to 3%. When the number of monomers increases, the degradation in the performance is even more severe for the equilibrium geometry. o For eight monomers, the CISD wavefunction recovers only half of the correlation energy and the Davidson correction remain more or less the same.

71 Dissociation of a water molecule Thick line: Full CI One the right: Difference between truncated CI and FCI Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

72 MRCISD wave functions in description of dissociation of a water molecule Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002) Difference between MRCISD and FCI

73 Perturbation theory In the variational method the starting point is to try to guess the wave function of the system. Perturbation theory is not variational. It is based on the idea that exact solutions for a system that resembles the true system are known. The solution for the real system is found by observing how the Hamiltonian for the two systems differ. In the picture (a) are shown the energy levels of harmonic oscillator. Its eigenfunctions and energy levels are known. In the picture (b) the system is perturbed by known potential energy term V which is the difference between potential energy curves of systems (a) and (b). The eigenfunctions and energy levels of system (b) are not known. The perturbation theory is devised to found approximate solutions for the properties of the perturbed system.

74 The Schrödinger equation for the perturbed state n is Hψ n = H 0 + λv ψ n = E n ψ n where λ is an arbitrary real parameter, V is a perturbation to the unperturbed Hamiltonian H 0, and subscript n = 1, 2, 3,... denotes different discrete states. The expressions produced by perturbation theory are not exact Accurate results can be obtained as long as the expansion parameter λ is very small. We expand ψ n and E n in Taylor series in powers of λ. The eigenvalue equation becomes H 0 + λv λ i ψ n i = λ i E n i λ i ψ n i i=0 i=0 i=0

75 Writing only the first terms we obtain H 0 + λv ψ n 0 + λψ n 1 = E n 0 + λe n 1 ψ n 0 + λψ n 1 The zeroth-order equation is simply the Schrödinger equation for the unperturbed system H 0 ψ n 0 = E n 0 ψ n 0 The first-order terms are those which are multiplied by λ H 0 ψ n 1 + Vψ n 0 = E n 0 ψ n 1 + E n 1 ψ n 0 When this is multiplied through by ψ n 0 from left and integrated, the first term on the left-hand side cancels with the first term on the right-hand side (The H 0 is hermitian). This leads to the firstorder energy shift: E n 1 = ψ n 0 V ψ n 0 This is simply the expectation value of the perturbation Hamiltonian while the system is in the unperturbed state. The energy of the nth state up to the first order is thus E n 0 + E n 1.

76 Interpretation of the first order correction to energy: E n 1 = ψ n 0 V ψ n 0 The perturbation is applied, but we keep the system in the quantum state ψ n 0, which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by ψ n 0 V ψ n 0 The true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as ψ n 0. Further shifts are given by the second and higher order corrections to the energy.

77 To obtain the first-order correction to the energy eigenstate, we recall the expression derived earlier H 0 ψ n 1 + Vψ n 0 = E n 0 ψ n 1 + E n 1 ψ n 0 and multiply it by ψ m 0, m n from left and integrate. We obtain E m 0 E n 0 ψ m 0 ψ n 1 = ψ m 0 V ψ n 0 We expand ψ n 1 as ψ n 1 = When two above equations are combined, we obtain ψ n 1 = m n m a m ψ m 0 ψ m 0 V ψ n 0 E n 0 E m 0 The first-order change in the n-th energy eigenfunction has a contribution from each of the energy eigenstates m n. ψ m 0

78 The second order correction to the energy is E n 2 = m n ψ m 0 V ψ n 0 E n 0 E m 0 2 Conclusion: Each term is proportional to the matrix element ψ m 0 V ψ n 0 This is a measure of how much the perturbation mixes state ψ n 0 with state ψ m 0 It is also inversely proportional to the energy difference between states ψ n 0 and ψ m 0, which means that the perturbation deforms the state to a greater extent if there are more states at nearby energies. Expression is singular if any of these states have the same energy as state ψ n 0, which is why we assumed that there is no degeneracy Higher-order deviations can be found by a similar procedure

79 Example: PT calculation for Helium atom Helium atom Hamiltonian H = = h r 1 r 2 r 1 + h It is convinient to choose the unperturbed system a a two-electron atom in which the electrons do not interact. The zeroth_order Hamiltonian then is r 12 H 0 = h 1 + h 2 The zeroth-order eigenfunctions have the hydrogen atom form ψ 0 = 1s 1 1s 2 = 1 π Z a e Z r 1+r 2 a 0 And the zeroth-order energy is simply E 0 = Z 2 = 4.0

80 The first-order perturbation-energy correction is E 1 = 1s 1 1s 2 1 r 12 1s 1 1s 2 = 5 8 Z = 5 4 Thus, the total energy of the helium atom (to the first order) is E = E 0 + E 1 = = 2.75 This result is equal to the single configurational energy in previous CI calculation. Thus, electron correlation does not seem to contribute to the first order energy correction. The total energy up to the second order is E = E 0 + E 1 + E 2 = = which is lower than true ground state energy

81 Møller Plesset perturbation theory Introduction: The Møller Plesset perturbation theory (MP) was published as early as 1934 by Christian Møller and Milton S. Plesset. The starting point is eigenfunction of the Fock-operator. It improves on the Hartree Fock method by adding electron correlation effects. MP theory is not variational. Calculated energy may be lower than true ground state energy. MP methods (MP2, MP3, MP4,...) are implemented in many computational chemistry codes. Higher level MP calculations, generally only MP5, are possible in some codes. However, they are rarely used because of their costs.

82 The MP-energy corrections are obtained with the perturbation V, which is defined as a difference between the true nonrelativistic Hamiltonian H and the sum of one-electron Fock operators F V = H F The Slater determinant ψ is the eigenfunction of the Fock-operator F Fψ = where ε i is the orbital energy belonging to the doubly occupied space orbital. The sum of MP zeroth-order energy and first order energy correction is (0) (1) E MP + EMP = ψ F ψ + ψ V ψ = ψ H ψ But ψ H ψ is the variational integral for the Hartree-Fock wave function ψ and it therefore equals the Hartree-Fock energy E HF. i ε i ψ (0) (1) E MP + EMP = EHF

83 In order to obtain the MP2 formula for a closed-shell molecule, the second-order correction formula is written on basis of doubly-excited Slater determinants (singly-excited Slater determinants vanish). (2) E MP = i>j a>b φ a 1 φ b 2 r 12 1 φ i 1 φ j 2 φ a 1 φ b 2 r 12 1 φ j 1 φ i 2 2 ε i + ε j ε a ε b where φ i and φ j are occupied orbitals and φ a and φ b are virtual (unoccupied) orbitals. The quantities ε i, ε j, ε a, and ε b are the corresponding orbital energies. Up to the second-order, the total electronic energy is given by the Hartree Fock energy plus second-order MP correction: (2) E = E HF + E MP

84 Calculated and experimental atomization energies (kj/mol) Molecule HF MP2 Exp. F H HF H 2 O O CO C 2 H CH Accuracy of the MP2 is satisfactory despite its relatively low computational cost MP2 usually overestimates bond energies

85 Calculated and experimental reaction enthalpies (kj/mol) Reaction HF MP2 Exp. CO + H 2 CH 2 O H 2 O + F 2 HOF + HF N 2 +3H 2 2NH C 2 H 2 +H 2 C 2 H CO 2 +4H 2 CH 4 + 2H 2 O CH 2 C 2 H O 3 +3H 2 3H 2 O The accuracy of MP2 is much improved compared to HF It is problematic to improve MPPT calculations systematically

86 Dissociation of a water molecule Thick line: FCI Full line: RHF reference state Dashed line: UHF reference state One the right: Difference between MPPT and FCI Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

87 Concluding remarks Systematic studies of MP perturbation theory have shown that it is not necessarily a convergent theory at high orders. The convergence properties can be slow, rapid, oscillatory, regular, highly erratic or simply non-existent, depending on the precise chemical system or basis set. Various important molecular properties calculated at MP3 and MP4 level are in no way better than their MP2 counterparts, even for small molecules. For open shell molecules, MPn-theory can directly be applied only to unrestricted Hartree Fock reference functions. However, the resulting energies often suffer from severe spin contamination, leading to very wrong results. A much better alternative is to use one of the MP2-like methods based on restricted open-shell Hartree Fock references.

88 Coupled cluster method First observations: Coupled cluster (CC) method, especially The CCSD(T), has become the gold-standard of quantum chemistry. CC theory was poised to describe essentially all the quantities of interest in chemistry, and has now been shown numerically to offer the most predictive, widely applicable results in the field. The computational cost is very high. So, in practice, it is limited to relatively small systems. Some facts: Coupled cluster (CC) is a numerical technique used for describing many-body systems. It starts from the Hartree-Fock molecular orbital method and adds a correction term to take into account electron correlation.

89 Some history: The CC method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics. In 1966 Jiri Cek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. Kümmel comments: I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost gave up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Jiri s work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then.

90 The wavefunction of the coupled-cluster theory is written in terms of exponential functions: ψ = e T φ 0 Where is a Slater determinant usually constructed from Hartree Fock molecular orbitals. The operator T is an excitation operator which, when acting on φ 0, produces a linear combination of excited Slater determinants. The exponential approach guarantees the size extensivity of the solution. For two subsystems A and B and corresponding excitation operators T A and T B, the exponential function admits for the simple factorization e T A+ T B = e T Ae T B. Therefore, aside from other advantages, the CC method maintains the property of size consistency. The cluster operator is written in the form T = T 1 + T 2 + T 3 + where T 1 is the operator of all single excitations, T 2 is the operator of all double excitations and so forth.

91 The exponential operator e T may be expanded into Taylor series: e T = 1 + T + T2 2! + = 1 + T 1 + T 2 + T T 1 T 2 + T In practice the expansion of T into individual excitation operators is terminated at the second or slightly higher level of excitation. Slater determinants excited more than n times contribute to the wave function because of the non-linear nature of the exponential function. Therefore, coupled cluster terminated at T n usually recovers more correlation energy than CI with maximum n excitations. A drawback of the method is that it is not variational which for truncated cluster expansion becomes where χ and ο are different functions E φ = φ e T He T φ φ φ E Φ = χ H ο φ φ 2

92 The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of T. The abbreviations for coupled-cluster methods usually begin with the letters CC (for coupled cluster) followed by S - for single excitations (shortened to singles in coupled-cluster terminology) D - for double excitations (doubles) T - for triple excitations (triples) Q - for quadruple excitations (quadruples) Thus, the operator in CCSDT has the form: T = T 1 + T 2 + T 3 Terms in round brackets indicate that these terms are calculated based on perturbation theory. For example, a CCSD(T) approach simply means: It includes singles and doubles fully Triples are calculated with perturbation theory.

93 Calculated and experimental atomization energies (kj/mol) Molecule HF CCSD MP2 Exp. F H HF H 2 O O CO C 2 H CH CCSD calculations produce qualitatively correct result Eventhought CCSD is expensive method, it is unfortunately not very accurate

94 Calculated and experimental reaction enthalpies (kj/mol) Reaction HF CCSD MP2 Exp. CO + H 2 CH 2 O H 2 O + F 2 HOF + HF N 2 +3H 2 2NH C 2 H 2 +H 2 C 2 H CO 2 +4H 2 CH 4 + 2H 2 O CH 2 C 2 H O 3 +3H 2 3H 2 O CCSD recovers majority of the electron correlation energy CCSD calculations do not achieve chemical accuracy (4 kj/mol)

95 Deviation of CI and CC energies from non-relativistic exact results (within B-O approximation) for H 2 O (mhartree = kj/mol) Method r e 1.5 r e 2 r e Hartree Fock CID CISD CISDT CISDTQ CCD CCSD CCSDT CCSDTQ CI converges (too) slowly to exact energy CC has superior performance but show fluctuations

96 We have learned that: Ab initio methods based on coupled cluster (CC) approach, are currently the most precise tool to calculate electron correlation effects. Some of the most accurate calculations for small to medium sized molecules use this method. The computational cost is very high. So, in practice, it is limited to relatively small systems. A drawback of the method is that it is not variational Unfortunately, for high precision work, the CCSD model is usually not accurate enough and CCSDT model is too expensive.

97 The solution: We can combine the CC and perturbation theory. Let the PT take care of the computationally expensive high excitation terms. The acronym for the most popular hybrid method is CCSD(T) where T in the brackets means perturbative triple excitations. The CCSD(T) method has become the gold-standard of quantum chemistry as it very reliably reaches the so-called chemical accuracy (energies within about 1-2 kcal/mol (1 kcal = 4.18 kj) of experimental values) when computing molecular properties for wide range of chemical species. Example: Water molecule CCSD(T) calculations For OH bond distances less than 3.5Å, the CCSD(T) works well, giving about 90% of the full CCSDT triples correction. The model breaks down at larger OH bond distances. The unrestricted CCSD(T) (based on UHF reference) does not provide good description of the dissociation process.

98 Calculated and experimental electronic atomization energies (kj/mol) Molecule HF CCSD CCSD(T) Exp. F H HF H 2 O O CO C 2 H CH CCSD(T) calculations produce accurate results Only for most problematic systems (such as ozone) higher order corrections are desirable

99 Calculated and experimental electronic reaction enthalpies (kj/mol) Reaction CCSD CCSD(T) Exp. CO + H 2 CH 2 O H 2 O + F 2 HOF + HF N 2 +3H 2 2NH C 2 H 2 +H 2 C 2 H CO 2 +4H 2 CH 4 + 2H 2 O CH 2 C 2 H O 3 +3H 2 3H 2 O CCSD(T) is generally an improvement over CCSD CCSD(T) models chemical reactions mostly within chemical accuracy (4 kj/mol)

100 Error in the reaction enthalpies (kj/mol) for 14 reactions involving small main-group element molecules

101 Dissociation of a water molecule Full line: RHF reference state, dashed line: UHF reference state Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

102 Comparison of models by the deviation from experimental molecular geometries of 29 small main-group element species cc-pvdz cc-pvtz

103 Difference to the FCI energy of various CC and MP levels of theory. Water molecule in equilibrium and stretched geometries.

104 Relationship between the calculated bond distances for the standard models (in pm)

105 Performance vs. accuracy of different ab initio methods Average errors in correlation energies compared to full CI applied to HB, HF, and H 2 O at both equilibrium and bond-stretched geometries Level of theory Equilibrium geometry Equilibrium and stretched geometry Scaling MP N 5 MP N 6 CISD N 6 CCD N 6 CCSD N 6 QCISD N 6 MP N 7 CCSD(T) N 7 QCISD(T) N 7 CCSDT N 8 CCSDTQ N 10

106 Basis sets A basis set is a set of functions used to create the molecular orbitals, which are expanded as a linear combination with coefficients to be determined. Usually these functions are centered on atoms, but functions centered in bonds or lone pairs have been used. Additionally, basis sets composed of sets of plane waves are often used, especially in calculations involving systems with periodic boundary conditions (continuous systems, surfaces). Quantum chemical calculations are typically performed within a finite set of basis functions. These basis functions are usually not the exact atomic orbitals, like the hydrogen atom eigenfunctions. If the finite basis is expanded towards an infinite complete set of functions, calculations using such a basis set are said to approach the basis set limit.

107 In the early days of quantum chemistry so-called Slater type orbitals (STOs) were used as basis functions due to their similarity with the eigenfunctions of the hydrogen atom. Their general definition is ψ nlm r, θ, φ = Nr n 1 e ζr a 0Y m l θ, φ where n = 1,2, is related to hydrogen atom principal quantum number, and l and m are related to hydrogen atom angular momentum and magnetic quantum numbers, respectively. N is a normalization factor, ζ is the effective nuclear charge, and Y l m θ, φ being the spherical harmonics. STOs have an advantage in that they have direct physical interpretation and thus are naturally good basis for molecular orbitals. From a computational point of view the STOs have the severe shortcoming that most of the required integrals needed in the course of the SCF procedure must be calculated numerically which drastically decreases the speed of a computation. Still, today there exist some modern and efficient computational chemistry program packages that use STOs (ADF).

108 STOs can be approximated as linear combinations of Gaussian type orbitals, which are defined as g ijk r = N x R i j x y R y z Rz k e α r R 2 N is a normalization factor, R is the atomic center, and α is an orbital exponent of the Gaussian function, respectively. GTOs are not really orbitals, they are simpler functions (Gaussian primitives). GTOs are usually obtained from quantum calculations on atoms (i.e. Hartree-Fock or Hartree-Fock plus some correlated calculations, e.g. CI). Typically, the exponents α are varied until the lowest total energy of the atom is achieved. For molecular calculations, certain linear combinations of GTOs will be used as basis functions. Such a basis function (contraction) will have its coefficients and exponents fixed. For example: φ 1 = ag 1 + bg 2 + cg 3 Where coefficients a, b, and c and the exponents α in functions g are fixed (i.e. are not variables).

109 The main difference to the STOs is that the variable r in the exponential function e α r R 2 is squared. Generally the inaccuracy at the center or the qualitatively different behaviour at long distances from the center have a marked influence on the results. The radial parts of the orbitals plotted in the figures

110 To understand why integrals over GTOs can be carried out when analogous STO-based integrals are much more difficult, one must consider orbital products ψ a, ψ b, ψ c, and ψ d where a, b, c, and d refer to different atomic centers. These products give rise to multi-center integrals, which often arise in polyatomic-molecule calculations, and which can not be efficiently performed when STOs are employed. For orbitals in the GTO form, can be rewritten as e α a r R a 2 e α c r R c 2 = e α a+α c r R 2 e α R a R c 2 where R = α ar a + α c R c α a + α c and α = α aα c α a + α c Thus, the product of two GTOs on different centers is equal to a single other GTO at center R between the original centers. As a result, even a four-center integral over GTOs can be written as two-center twoelectron integral. A similar reduction does not arise for STOs.

111 In GTOs N x R x i y R y j z Rz k e α r R 2 the sum of the exponents of the cartesian coordinates, L = i + j + k, is used to mark functions as s-type (L = 0), p-type (L = 1), d-type (L = 2), and so on Unfortunately GTOs are not eigenfunctions of the squared angular momentum operator L 2. However, combinations of GTOs are able to approximate correct nodal properties of atomic orbitals by taking them with different signs. For example combining three d-type cartesian GTOs yields a cartesian GTO of s-type: g g g 002 g 000 Today, there are hundreds of basis sets composed of GTOs. The smallest of these are called minimal basis sets, and they are typically composed of the minimum number of basis functions required to represent all of the electrons on each atom. The largest of these can contain literally dozens to hundreds of basis functions on each atom.

112 A minimum basis set is one in which a single basis function is used for each orbital in a Hartree-Fock calculation on the free atom. However, for atoms such as lithium, basis functions of p-type are added to the basis functions corresponding to the 1s and 2s orbitals of the free atom. For example, each atom in the first row of the periodic system (Li - Ne) would have a basis set of five functions (two s-type functions and three p-type functions). The most common minimal basis set is STO-nG, where n is an integer. This n value represents the number GTOs used to approximate STO for both core and valence orbitals. Minimal basis sets typically give rough results that are insufficient for research-quality publication, but are much cheaper than their larger counterparts. Commonly used minimal basis sets of this type are: STO-3G, STO-4G, STO-6G

113 The minimal basis sets are not flexible enough for accurate representation of orbitals Solution: Use multiple functions to represent each orbital For example, the double-zeta basis set allows us to treat each orbital separately when we conduct the Hartree-Fock calculation. ψ 2s r = c 1 ψ 2s r; ζ 1 + c 2 ψ 2s r; ζ 2 where 2s atomic orbital is expressed as the sum of two STOs. The ζ-coefficients account for how large the orbitals are. The constants c 1 and c 2 determine how much each STO will count towards the final orbital.

114 The triple and quadruple-zeta basis sets work the same way, except use three and four Slater equations (linear combination of GTOs) instead of two. The typical trade-off applies here as well, better accuracy...more time/work. There are several different types of extended basis sets split-valence polarized sets diffuse sets correlation consistent sets

115 Pople s split-valence basis sets n-ijg or n-ijkg. n - number of GTOs for the inner shell orbitals; ij or ijk number of GTOs for basis functions in the valence shell. The ij notations describes sets of valence double zeta quality and ijk sets of valence triple zeta quality. The s-type and p-type functions belonging to the same electron shell are folded into a sp-shell. In this case, number of s-type and p-type GTOs is the same, and they have identical exponents. However, the coefficients for s-type and p-type basis functions are different.

116 Example: Four s-type GTOs used to represent 1s orbital of hydrogen as: ψ 1s = N 1 e r N 2 e r N 3 e r N 4 e r2 where N i is a normalization constant for a given GTO. These GTOs may be grouped in 2 basis functions. The first basis function contains only 1 GTO: 3 GTOs are present in the second basis function: φ 1 = N 1 e r2 φ 2 = N N 2 e r N 3 e r N 4 e r2 where N is a normalization constant for the whole basis function. In this case, 4 GTOs were contracted to 2 basis functions. It is frequently denoted as 4s 2s contraction. The coefficients in function are then fixed in subsequent molecular calculations.

117 Example: Silicon 6-31G basis set The corresponding exponents for s-type and ptype basis functions are equal but coefficients in s-type and p-type basis functions are different. GTOs are normalized here since coefficients for basis functions consisting of one GTO (last row) are exactly 1. The basis set above represents the following contraction 16s, 10p 4s, 3p

118 Example: 3-21G basis set of carbon Surface and contour plot of p type basis function including two Gaussians. α sp,1 : = φ 2py : = y c 2p,1 φ 2p,1 + c 2p,2 φ 2p,2 α sp, φ 2p,1 : = 2 α sp,1 π 3 4 e α sp,1 x 2 +y 2 +z 2 c 2p,1 : = c 2p,2 : =

119 Polarized basis sets Polarization functions denoted in Pople s sets by an asterisk Two asterisks, indicate that polarization functions are also added to light atoms (hydrogen and helium). Polarization functions have one additional node. For example, the only basis function located on a hydrogen atom in a minimal basis set would be a function approximating the 1s atomic orbital. When polarization is added to this basis set, a p-type function is also added to the basis set. 6-31G**

120 Polarization functions add flexibility within the basis set, effectively allowing molecular orbitals to be more asymmetric about the nucleus. This is an important for accurate description of bonding between atoms, because the precence of the other atom distorts the environment of the electrons and removes its spherical symmetry.

121 Similarly, d-type functions can be added to a basis set with valence p-type orbitals, and so on. High angular momentum polarization functions (d, f, ) are important for heavy atoms

122 Some observations concerning polarization functions: The exponents for polarization functions cannot be derived from Hartree-Fock calculations for the atom, since they are not populated. In practice, these exponents are estimated using well established rules of thumb or by using a test set of molecules. The polarization functions are important for reproducing chemical bonding. They should be included in all calculations where electron correlation is important. Adding them is costly. Augmenting basis set with d-type polarization functions adds 5 basis function on each atom while adding f-type functions adds 7.

123 The basis sets are also frequently augmented with the so-called diffuse functions. These Gaussian functions have very small exponents and decay slowly with distance from the nucleus. Diffuse gaussians are usually of s-type and p-type. Diffuse functions are necessary for correct description of anions and weak bonds (e.g. hydrogen bonds) and are frequently used for calculations of properties (e.g. dipole moments, polarizabilities, etc.). For the Pople s basis sets the following notaton is used: n-ij+g, or n-ijk+g when 1 diffuse s-type and p-type gaussian with the same exponents are added to a standard basis set on heavy atoms. The n-ij++g, or n-ijk++g are obtained by adding 1 diffuse s-type and p-type gaussian on heavy atoms and 1 diffuse s-type gaussian on hydrogens.

124 Diffuse functions The area which is modelled by diffuse functions. Diffuse functions are very shallow Gaussian basis functions, which more accurately represent the tail portion of the atomic orbitals, which are distant from the atomic nuclei.

125 Correlation consistent basis sets are widely used basis sets are those developed by Dunning and co. These basis sets have become the current state of the art for correlated calculations Designed to converge systematically to the complete basis set (CBS) limit using extrapolation techniques For first- and second-row atoms, the basis sets are cc-pvnz where n=d,t,q,5,6,... (D=doublezeta, T=triple-zeta, etc.) The cc-p, stands for correlation consistent polarized and the V indicates they are valence only basis sets. They include successively larger shells of polarization (correlating) functions (d, f, g, etc.). The prefix aug means that the basis is augmented with diffuse functions Examples: cc-pvtz, aug-cc-pvdz, aug-cc-pcv5z

126 The complete basis set limit (CBS) can be approximately approached by extrapolation techniques

127 Correlation consistent basis sets are built up by adding shells of functions to a core set of atomic Hartree-Fock functions. Each function in a shell contributes very similar amounts of correlation energy in an atomic calculation. For the 1st and 2nd row atoms, the cc-pvdz basis set adds 1s, 1p, and 1d function. The cc-pvtz set adds another s, p, d, and f function, etc. For third-row atoms, additional functions are necessary; these are the cc-pv(n+d)z basis sets.

128 cc-pvdz for carbon **** C 0 S S S P P D **** cc-pvtz for carbon **** C 0 S S S S P P P D D F ****

129 Basis set errors of total energies for several first- and second-row atoms. Tier-n denotes different tiers of the FHI-aims-2009 basis sets and NAO-VCC-nZ denotes numerically tabulated atom-centered orbital basis sets with valence-correlation consistency.

130 Basis set superposition error Calculations of interaction energies are susceptible to basis set superposition error (BSSE) if they use finite basis sets. As the atoms of interacting molecules or two molecules approach one another, their basis functions overlap. Each monomer borrows functions from other nearby components, effectively increasing its basis set and improving the calculation. The counterpoise approach (CP) calculates the BSSE employing ghost orbitals. In the uncorrected calculation of a dimer AB, the dimer basis set is the union of the two monomer basis sets. The uncorrected interaction energy is V AB G = E AB G, AB E A A E B B where G denotes the coordinates that specify the geometry of the dimer and E AB G, AB the total energy of the dimer AB calculated with the full basis set AB of the dimer at that geometry. Similarly, E A A and E B B denote the total energies of the monomers A and B, each calculated with the appropriate monomer basis sets A and B, respectively.

131 The counterpoise corrected interaction energy is V cc AB G = E AB G, AB E A G, AB E B G, AB where E A G, AB and E B G, AB denote the total energies of monomers A and B, respectively, computed with the dimer basis set at geometry G, i.e. in the calculation of monomer A the basis set of the other monomer B is present at the same location as in dimer A, but the nuclei of B are not. In this way, the basis set for each monomer is extended by the functions of the other monomer. The counterpoise corrected energy is thus The counterpoise correction provides only an estimate of the BSSE. BSSE is present in all molecular calculations involving finite basis sets but in practice its effect is important in calculations involving weakly bound complexes. Usually its magnitude is few kj/mol to binding energies which is often very significant.

132 The frozen core approximation The lowest-lying molecular orbitals are constrained to remain doubly-occupied in all configurations. The lowest-lying molecular orbitals are primarily these inner-shell atomic orbitals (or linear combinations thereof). The frozen core for atoms lithium to neon typically consists of the 1s atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals 1s, 2s, and 2p. A justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than are the valence electrons. The error introduced by freezing the core orbitals is nearly constant for molecules containing the same types of atoms. In fact, it is sometimes recommended that one employ the frozen core approximation as a general rule because most of the basis sets commonly used in quantum chemical calculations do not provide sufficient flexibility in the core region to accurately describe the correlation of the core electrons.

133 The pseudopotentials The pseudopotential is an attempt to replace the complicated effects of the motion of the core electrons and nucleus with an effective core potential (ECP) Motivation: Reduction of basis set size Reduction of number of electrons Inclusion of relativistic and other effects Approximations: Pseudopotentials imply the frozen core approximation Valence-only electrons Assumes that there is no significant overlap between core and valence WF

134 Towards exact solution of Scrödinger equation

135 Multi reference methods Multi-configurational self-consistent field (MCSCF) MCSCF is a method to generate qualitatively correct reference states of molecules in cases where Hartree Fock is not adequate It uses a linear combination of configuration state functions (CSF) or Slater determinants to approximate the exact electronic wavefunction It can be considered a combination between configuration interaction (where the molecular orbitals are not varied but the expansion of the wave function) and Hartree-Fock (where there is only one determinant but the molecular orbitals are varied). MCSCF method is an attempt to generalize the Hartree Fock model and to treat real chemical processes, where nondynamic correlation is important, while keeping the conceptual simplicity of the HF model as much as possible. Although MCSCF itself does not include dynamic correlations, it provides a good starting point for such studies.

136 Example: Hydrogen molecule MCSCF treatment HF gives a reasonable description of H 2 around the equilibrium geometry About for the bond length compared to a experimental value 84 kcal/mol for the bond energy (exp. 109 kcal/mol). Problem: At large separations the presence of ionic terms H + + H (which have different energy than H + H) lead to an unphysical solution. Solution: The total wave function of hydrogen molecule (including hydrogens A and B) can be written as a linear combination of configurations built from bonding and anti-bonding orbitals ψ MC = C 1 ψ 1 + C 2 ψ 2 where ψ 1 is the bonding orbital 1s A + 1s B and ψ 2 is the anti-bonding orbital 1s A 1s B. In this multi configurational description of the H 2 chemical bond, C 1 = 1 and C 2 = 0 close to equilibrium, and C 1 will be comparable to C 2 for large separations.

137 In the complete active space CASSCF method the occupied orbital space is divided into a sets of active, inactive and secondary orbitals. The active space orbitals are highlighted in yellow, while inactive and secondary orbitals are greyed out. It allows complete distribution of active (valence) electrons in all possible ways. Corresponds to a FCI in the active space. The orbitals not incorporated in the active space remain either doubly occupied (inactive or core space) or empty (secondary). The restricted active space method (RASSCF) uses only selected subspaces of active orbitals.it could for example restrict the number of electrons to at most 2 in some subset.

138 For instance, the description of a double bond isomerization requires an active space including all π-electrons and π-orbitals of the conjugated system incorporating the reactive double bond. This choice is motivated by the need to allow for all possible variations in the overlap between the set of p-orbitals forming the reacting π-system along the reaction coordinate. More generally, the selection of the active space electrons and orbitals is a chemical problem, and often is not a straightforward one. It is often a challenge to generate configuration space sufficiently flexible to describe the physical process and yet sufficiently small to be computationally tractable. Reaction schemes considered for the isomerization of the propene radical cation.

139 Starts with a CASSCF calculation, which describes the static electron correlation by including the nearly degenerate electron configurations in the wavefunction. The dynamical electron correlation is included by substitutions of occupied orbitals by virtual orbitals in the individual configuration state functions (CSFs). The truncation of the expansion space to single and double substitutions is usually mandated by the very steep increase in the number of CSFs, and the consequent computational effort. Multi reference configuration interaction (MRCI) The disadvantage of truncated CI is its lack of sizeextensivity. A correction of this problem is by no means as straightforward in MRCI as in the single reference case. A representation of the configurations included in the MRCI wavefunction taking one of the CASSCF configurations as the reference.

140 The multi-reference perturbation theory (MR-PT) is the most cost-effective multi-reference approach compared to multi-reference CI (MRCI) and multi-reference Coupled Cluster (MR-CC). Example: In computational photochemistry and photobiology the CASPT2 method leads often to suitably accurate vertical excitation energies (with errors within 3 kcal/mol with respect to observation). This method has been shown to benefit from a balanced cancellation of errors. While the CASPT2 has been successfully applied in many studies, it typically requires experienced users that are familiar with its pitfalls. In fact, this requirement is generally true for multi-reference methods, such as CASSCF and MRCI, which require users capable of selecting variables such as the active space and the number of states to include in the calculation (i.e. they are not black-box methods). MR-CC is approximately size-extensive but in general not clearly as advantageous compared to MR- CI as in single reference picture

141 Explicitly correlated methods The electron-electron distance r 12 ought to be included into the wavefunction if highly accurate computational results were to be obtained. Example: For a helium atom we can choose the trial function as follows: φ r 1, r 2, r 12 = C 1 + pr 12 + q r 1 r 2 2 exp α r 1 + r 2 where C is normalization constant and p, q, and α are variational parameters. After optimization: p = 0.30, q = 0.13, and α = E = (Three parameters) E = (1024 parameters) E = (Experimental value)

142 Explicitly correlated methods R12 and F12 methods include two-electron basis functions that depend on the interparticle distance r 12 These theories bypass the slow convergence of conventional methods High accuracy can be achieved dramatically faster With these methods it has been achieved kj/mol accuracy for molecular systems consisting of up to 18 atoms This result is well below the so called chemical accuracy, that is, an error of 1 kcal/mol (4.184 kj/mol) Methods: CCSD-R12 or F12, CCSD(T)-R12 or F12, MP2-R12 or F12

143 Example: Calculated and experimental geometric parameters Molecule H 2 O CCSD(T)-F12a/ VDZ-F12 CCSD(T)/ AVDZ CCSD(T)/ AV6Z Exp. R θ H 2 S R θ NH 3 R θ CCSD(T)-F12a is in impressive agreement with experiment Computational cost of CCSD(T)-F12a calculations are reduced because of negligible BSSE and small basis set

144 Linear scaling approaches Computational expense can be reduced by simplification schemes. The local approximation. Interactions of distant pairs of localized molecular orbitals are neglected in the correlation calculation. This sharply reduces the scaling with molecular size, a major problem in the treatment of biologically-sized molecules. Methods employing this scheme are denoted by the prefix L, e.g. LMP2. Scaling is reduced to N. The density fitting scheme The charge densities in the multi-center integrals are treated in a simplified way. This reduces the scaling with respect to basis set size. Methods employing this scheme are denoted by the prefix df-, for example the density fitting MP2 is df-mp2 (lower-case is advisable to prevent confusion with DFT). Both schemes can be employed together For example, as in the recently developed df-lmp2 and df-lccsd methods. In fact, df-lmp2 calculations are faster than df-hartree Fock calculations and thus are feasible in nearly all situations in which also DFT is. Compared to local methods (LMP2 and LCCSD) these methods are ten times faster.

145 Example: The interaction energy between two benzenes

146 Density functional theory Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function, which in this case is the electron density. The multiple determinant calculations (CI for example) require very large basis sets due to the poor convergence of the correlation energy when the inter-electronic distance becomes very small. However, DFT can produce accurate results with relatively small basis sets. DFT has become the most popular and versatile method in computational chemistry, accounting for approximately 90% of all calculations today. The reason for this preference is that DFT scales with the same order as HF theory (N 3, where N is proportional to system size)

147 DFT avoids the expense of the more traditional methods, deriving the energy directly from the electron probability density, rather than the molecular wavefunction, thus drastically reducing the dimensionality of the problem. Regardless of how many electrons one has in the system, the density is always 3 dimensional. Some history: DFT has been very popular for calculations in solid state physics since the 1970s. DFT calculations agreed quite satisfactorily with experimental data. Also, the computational costs were relatively low when compared to Hartree-Fock theory and its descendants. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. DFT is now a leading method for electronic structure calculations in chemistry and solid-state physics.

148 Despite the improvements in DFT, there are still difficulties in using density functional theory to properly describe intermolecular interactions, especially: van der Waals forces (dispersion) charge transfer excitations transition states global potential energy surfaces and some other strongly correlated systems calculations of the band gap in semiconductors. Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g., interacting noble gas atoms) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.

149 Electron density Define ρ 1 r = N ψ r 1, r 2,, r N 2 dσ 1 dr 2 dr N ρ r = ρ 1 r dσ 1 o ρ 1 describes the probability of finding any of the N electrons within the volume element in the spin state σ, with the other N-1 electrons having arbitrary positions and spin states o ρ is an observable (e.g. X-ray spectroscopy) Properties ρ r = 0 ρ r dr 1 = N

150 Pair density Let s generalize: the probability for finding two electrons in spin states and in the volume elements and is given by the pair density ρ 2 r 1, r 2 = N N 1 ψ r 1, r 2,, r N 2 dσ 1 dσ 2 dr 3 dr N P r 1, r 2 = ρ 2 r 1, r 2 dσ 1 dσ 2 The pair density contains all information about electron correlation, and we can express the energy of any system in any state as E = ρ r 1 dr 1 + i ρ r 1 r 1 R i dr P r 1, r 2 r 1 r 2 dr 1 dr 2

151 Rewrite ρ 2 r 1, r 2 = ρ 1 r 1 ρ 1 r f r 1, r 2 Where f r 1, r 2 is called correlation factor. For example f = 0 corresponds to uncorrelated case. The difference between the probability to find any electron in dr 2 while there is an electron in dr 1 and uncorrelated case is called the exchange correlation hole We observe that h xc r 1 ; r 2 = ρ 2 r 1, r 2 ρ 1 r 1 ρ 1 r 2 = ρ 1 r 2 f r 1, r 2 E ee = 1 2 P r 1, r 2 r 1 r 2 dr 1 dr 2 = 1 2 ρ r 1 ρ r 2 r 1 r 2 dr 1 dr ρ r 1 h xc r 1 ; r 2 r 1 r 2 dr 1 dr 2 = J ρ + E xc ρ

152 The h xc can be formally splitted into h xc = h x σ 1 =σ 2 r 1 ; r 2 + h c r 1 ; r 2 where h x is exchange hole, due to Pauli principle (wave function antisymmetry) and h c is correlation hole, due to electrostatic repulsion. Only the h xc can be given proper meaning The Hartree-Fock theory accounts for h x but neglects h c.

153 The Hohenberg-Kohn theorems The Hohenberg-Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential Theorem 1. The external potential and hence the total energy, is a unique functional of the electron density The first H-K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density. This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states. Theorem 2. The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact ground state density The second Hohenberg-Kohn theorem has two drawbacks. Firstly, it assumes that there is no degeneracy in the ground state, and secondly the density has unknown form.

154 The uniform electron gas There is no systematic way to find or improve a density functional. The most appealing way forward is to find the exact solution for a model system, and then assume that the system of interest behaves similarly to the model. The first density functionals were due to Thomas, Fermi, and Dirac, all of which used the uniform electron gas as their model. The uniform electron gas is defined as a large number of electrons N in a cube of volume V, throughout which there is a uniform spread of positive charge sufficient to make the system neutral. The uniform gas is then defined as the limit N, V, with the density ρ = N/V remaining finite. Although it does bear some resemblance to electrons in metals, its widespread use is due to its simplicity: It is completely defined by one variable, the electron density ρ. Using the uniform electron gas, an expression for the kinetic energy (the Thomas-Fermi kinetic functional) can be derived T TF27 ρ σ = π ρ 3 σ r dr, where σ can take the values of α or β.

155 The importance of simple Thomas-Fermi model is not how well it performs in computing the ground state energy and density but more as an illustration that the energy can be determined purely using the electron density. When applied to atoms and molecules the Thomas-Fermi functional yields kinetic energies that are about 10% too small. Similarly, an expression for the exchange energy of the uniform electron gas can be calculated (the Dirac exchange functional) E x D30 ρ σ = π 1 3 ρ σ 4 3 r dr The Dirac functional also gives exchange energies that are roughly 10% smaller than those from HF theory.

156 The non-uniform electron gas The electron densities of atoms and molecules are often far from uniform, so functionals based on systems which include an inhomogeneous density should perform better. In 1935 von Weizsacker placed infinitesimally small ripples on the uniform electron gas and calculated the second order correction to the kinetic energy T W35 ρ σ = T TF27 ρ σ ρ σ 5 3 r x σ 2 r dr Where x r is a dimensionless quantity, the reduced density gradient x r = ρ r ρ 4 3 r Unfortunately the original derivation was flawed and the above functional is too large by a factor of nine. The corrected functional is a large improvement on T TF27 ρ σ, yielding kinetic energies typically within 1% of HF theory.

157 A similar correction was made to the Dirac exchange functional by Sham and Kleinman. The second order correction to the exchange energy is E x SK71 ρ σ = E x D30 ρ σ 5 36π 5 3 ρ σ 4 3 r x σ 2 r dr The corrected functional gives exchange energies that are typically within 3% of HF; however, it is not seen as an improvement over the Dirac functional, as the potential is unbounded in the Rydberg regions of atoms and molecules.

158 Kohn-Sham DFT The kinetic energy has a large contribution to the total energy. Therefore even the 1% error in the kinetic energy of the Thomas-Fermi-Weizsacker model prevented DFT from being used as a quantitative predictive tool. Thus DFT was largely ignored until 1965 when Kohn and Sham introduced a method which treated the majority of the kinetic energy exactly. Key idea: The intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The theory is based on the reference system: N noninteracting electrons moving in effective potential v eff, each in one of N orbitals, ψ i. The central equation in Kohn-Sham DFT is the one-electron Schrödingerlike equation expressed as: 1 2 i 2 +v eff r i ψ i = ε i ψ i

159 The kinetic energy and electron density are given by T s ρ = i ψ i 1 2 i 2 ψ i and the total energy is given by ρ r = i ψ i r 2 E ρ = T s ρ + E ee ρ + E ne ρ = T s ρ + J ρ + E xc ρ + E ne ρ Where E ne ρ is energy arising from electron-nuclear interaction

160 The KS equations are very similar to the Hartree Fock equations. Setting the exchange-correlation energy term to the HF exchange potential yields the HF equations. Just like the HF equations, the KS equations are solved iteratively. Differences: The KS orbitals are simply a way of representing the density; they are not (as in HF) an approximation of the wavefunction. HF theory is variational, providing an upper bound to the exact energy, yet DFT is only variational if the exact energy functional is used. Because the KS equations so closely follow the restricted HF equations, both the restricted open shell and unrestricted methodologies are readily available. However, the KS equations are formally exact (given the exact E xc ρ ), so it must be able to produce an excess of β electron density at points in the molecule, and therefore only the unrestricted formalism is appropriate.

161 Just as in HF theory, the KS equations are solved by expanding the orbitals over a basis set. The major advantage of DFT is that the basis set requirements are far more modest than the more conventional correlated methods In DFT the basis set only needs to represent the one electron density the inter-electron cusp is accounted for by the effective potential, v eff. In the more traditional methods the basis set describes the entire N-electron wavefunction, requiring an accurate description of the cusp which is sensitive to the basis set. The kinetic energy functional is known exactly. The exchange-correlation part of the total-energy functional remains unknown and must be approximated.

162 Local-density approximation In local-density approximation (LDA), the exchange-correlation energy functional E xc ρ depends only on the density at the coordinate where the functional is evaluated. E LDA xc ρ = ε xc ρ ρ r dr where ε xc ρ is the exchange-correlation energy density. The exchange-correlation energy is decomposed into exchange and correlation terms linearly: E xc = E x + E c so that separate expressions for E x and E c are sought. The uniform electron gas functional is used for the E x : E x ρ σ = π 1 3 ρ σ 4 3 r dr The correlation energy is more complicated and numerous different approximations exist for E c.

163 Strictly, the LDA is valid only for slowly varying densities. LDA works surprisingly well with calculations of atoms, molecules, and solids (especially for metals). o Systematic error cancelation: Typically, in inhomogeneous systems LDA underestimates correlation but overestimates exchange, resulting in unexpectedly good energy value. LDA tends to overestimate cohesive energies by 15-20% and underestimates lattice constants by 2-3% for metals and insulators. Problem with LDA becomes more severe for weakly bonded systems, such as vdw and H-bonded systems. o For example, the binding energy of the water dimer and the cohesive energy of bulk ice are both >50% too large with LDA compared to the experimental values. o Long range vdw interactions are completely missing in LDA.

164 Generalized gradient approximation LDA treats all systems as homogeneous. However, real systems are inhomogeneous. Generalized gradient approximation (GGA) takes this into account by including the derivative information of the density into the exchange-correlation functionals. E GGA xc ρ = f ρ r, ρ r dr It is not the physics per se but obtained results that guide the mathematical constructs o Some successful functionals are not based on any physical considerations o For example let s look two popular functionals: In PBE, the functional parameters are obtained from physical constraints (non-empirical). In B88, functional parameters are obtained from empirical fitting (empirical). GGAs are often called semi-local functionals due to their dependence on ρ r.

165 In comparison with LDA, GGA tend to improve total energies, atomization energies, energy barriers and structural energy differences. Especially for covalent bonds and weakly bonded systems many GGAs are far superior to LDA o Overall though because of flexibility of a choice of f ρ r, ρ r a zoo of GGA functionals have been developed and depending on the system under study a wide variety of results can be obtained. GGA expand and soften bonds, an effect that sometimes corrects and sometimes overcorrects the LDA prediction Whereas the ε xc ρ (in LDA) is well established, the best choice for f ρ r, ρ r is still a matter of debate

166 The hybrid functionals Q: Why bother with making GGA exchange functionals at all we know that the HF exchange is exact; i.e. These fourth generation functionals add exact exchange calculated from the HF functional to some conventional treatment of DFT exchange and correlation. LDA and GGA exchange and correlation functionals are mixed with a fraction of HF exchange The most widely used, particularly in the quantum chemistry community, is the B3LYP functional which employs three parameters, determined through fitting to experiment, to control the mixing of the HF exchange and density functional exchange and correlation.

167 Equilibrium C-C and C=C bond distances (Å) Molecule HF B3LYP MP2 Exp. But-1-yne-3-ene Propyne ,3-Butadiene Propene Cyclopropane Propane Cyclobutane Cyclopropene Allene Propene Cyclobutene But-1-yne-ene ,3-Butadiene Cyclopentadiene Mean error

168 The meta-ggas These are the third generation functionals and use the second derivative of the density, 2 ρ r and/or kinetic energy densities, τ r = 1 2 i φ ρ 2, as additional degrees of freedom. In gas phase studies of molecular properties meta-ggas have been shown to offer improved performance over LDAs and GGAs. Another class of functionals, known as hybrid meta-gga functionals, is combination of meta-gga and hybrid functionals with suitable parameters fitted to various molecular databases.

169 DFT summary In practice, DFT can be applied in several distinct ways depending on what is being investigated. In solid state calculations, the local density approximations are still commonly used along with plane wave basis sets, as an electron gas approach is more appropriate for electrons delocalised through an infinite solid. In molecular calculations more sophisticated functionals are needed, and a huge variety of exchangecorrelation functionals have been developed for chemical applications. In the chemistry community, one popular functional is known as BLYP (from the name Becke for the exchange part and Lee, Yang and Parr for the correlation part). Even more widely used is B3LYP which is a hybrid functional in which the exchange energy, in this case from Becke s exchange functional, is combined with the exact energy from Hartree Fock theory. The adjustable parameters in hybrid functionals are generally fitted to a training set of molecules. Unfortunately, although the results obtained with these functionals are usually sufficiently accurate for most applications, there is no systematic way of improving them (in contrast to methods like configuration interaction or coupled cluster theory) Hence in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiments.

170 Semiempirical methods Semiempirical methods of quantum chemistry start out from the ab initio formalism (HF-SCF) and then introduce assumptions to speed up the calculations, typically neglegting many of the less important terms in the ab initio equations. In order to compensate for the errors caused by these approximations, empirical parameters are incorporated into the formalism and calibrated against reliable experimental or theoretical reference data. It is generally recognized that ab initio methods (MP, CI, and CC) and even DFT can give the right result for the right reason, not only in principle, but often in practice, and that semiempirical calculations can offer qualitatively correct results of useful accuracy for many larger and chemically interesting systems. Semiempirical calculations are usually faster than DFT computations by more that two orders of magnitude, and therefore they often remain the method of choice in applications that involve really large molecules (biochemistry) or a large number of molecules or a large number of calculations (dynamics). Today, many chemical problems are solved by the combined use of ab initio, DFT, and semiempirical methods.

171 Basic concepts: A semiempirical model employs a Hartree-Fock SCF-MO treatment for the valence electrons with a minimal basis set. The core electrons are taken into a account through the effective nuclear charge, which is used in place of the actual nuclear charge to account for the electron-electron repulsions, or represented by ECP. Dynamic electron correlation effects are often included in an average sense. The standard HF equations are simplified by neglegting all three-center and four-center two electron integrals. One-center and two-center integral calculations are also simplified. For example CNDO (complete neglect of differential overlap), INDO (intermediate neglect of differential overlap), and NDDO (neglect of diatomic differential overlap) schemes differ how they introduce approximations in onecenter and two-center integral calculations.

172 Consider the following two-electron integral ψ i 1 ψ j 2 1 r 12 ψ j 2 ψ i 1 dτ 1 dτ 2 where ψ are expanded in terms of atom centered basis functions φ as usual ψ i = k c ik φ k Thus, the above integral includes terms of the following type kl rs = φ k 1 φ l 1 1 r 12 φ r 2 φ s 2 dτ 1 dτ 2 The zero differential overlap approximation ignores integrals that contain the products where k is not equal to l and r is not equal to s kl rs = kk rr total number of such integrals is reduced approximately from N 4 8 (Hartree Fock) to N 2 2.

173 The CNDO method use the zero differential overlap approximation completely. Spherically symmetric orbitals only Methods based on the intermediate neglect of differential overlap, such as INDO, do not apply the zero differential overlap approximation when all four basis functions are on the same atom One-centre repulsion integrals between different orbitals Methods that use the neglect of diatomic differential overlap, NDDO, do not apply the zero differential overlap approximation when the basis functions for the first electron are on the same atom and the basis functions for the second electron are on the same atom. Includes some directionality of orbitals The approximations work reasonably well when the integrals that remain are parametrized

174 The one-center and two-center integrals are determined directly from experimental data (onecenter integrals derived from atomic spectroscopic data) orcalculated using analytical formulas or represented by suitable parametric expressions (empirical or high-level ab initio). Most succesful semi-empirical methods (for studying ground-state potential energy surfaces) are based on NDDO scheme o In MNDO (Modified Neglect of Diatomic Overlap), the parametrisation is focused on ground state properties (heats of formation and geometries), ionization potentials, and dipole moments o Later, MNDO was essentially replaced by two new methods, PM3 and AM1, which are similar but have different parametrisation methods (more parameters and thus, more flexibility).

175 Applications: Large biomolecules with thousands of atoms o The accuracy of semiempirical methods is best for organic compounds Medicinal chemistry and drug design o Semiempirical methods are well suited for quantitative structure-property relationship and quantitative structure-activity relationship (QSPR and QSAR, respectively) modeling. Nanoparticles o Large fullerenes and nanotubes are prime examples Solids and surfaces o Large clusters which approach bulk limit Direct reaction dynamics o Thousands or even millions single point calculations Electronically excited states of large molecules and photochemistry. Alternative for TDDFT.

176

177 Graphical models In addition to numerical quantities (bond lengths and angles, energies, dipole moments, ) some chemically useful information is best displayed in the form of images. For example molecular orbitals, the electron density, electrostatic potential, etc. These objects can be displayed on screen using isosurface f x, y, z = constant The constant may be some physical observable of interest, for example, the size of the molecule.

178 Molecular Orbitals Chemists are familiar with the molecular orbitals of simple molecules. They recognize the σ and π orbitals of acetylene, and readily associate these with the molecule s σ and π bonds

179 A simple example where the shape of the highest occupied molecular orbital (HOMO) foretells of chemistry is found in cyanide anion. Cyanide acts as a nucleophile in SN2 reactions :N C: + CH 3 I :N C CH 3 + I The HOMO in cyanide is more concentrated on carbon (on the right) than on nitrogen suggesting, as is observed, that it will act as a carbon nucleophile.

180 Molecular orbitals do not even need to be occupied to be informative. For example, the lowest-unoccupied molecular orbital (LUMO) of perpendicular benzyl cation anticipates the charge delocalization. + It is into the LUMO, the energetically most accessible unfilled molecular orbital, that any further electrons will go. Hence, it may be thought of as demarking the location of positive charge in a molecule. Examination of the LUMO of methyl iodide helps to rationalize why iodide leaves following attack by cyanide. This orbital is antibonding between carbon and iodine (there is a node in the bonding region), meaning that donation of the electron pair from cyanide will cause the CI bond to weaken and eventually break.

181 Woodward and Hoffmann first introduced organic chemists to the idea that so-called frontier orbitals (the HOMO and LUMO), which often provide the key to understanding why some chemical reactions proceed easily whereas others do not HOMO in cis-1,3-butadiene is able to interact favorably with the LUMO in ethylene (constructive overlap) to form cyclohexene Interaction between the HOMO on one ethylene and the LUMO on another ethylene is not favorable, and concerted addition to form cyclobutane would not be expected

182 Electron density Isodensity surfaces may either serve to locate atoms, delineate chemical bonds, or to indicate overall molecular size and shape. The regions of highest electron density surround the heavy (non-hydrogen) atoms in a molecule. Thus, the X-ray diffraction experiment locates atoms by identifying regions of high electron density. Also interesting, are regions of slightly lower electron density. For example, isodensity surface (0.1 electrons/au 3 ) for cyclohexanone conveys essentially the same information as a conventional skeletal structure model, that is, it depicts the locations of bonds

183 A low density surface (0.002 electrons/au 3 ), serves to portray overall molecular size and shape. This is, of course, the same information portrayed by a conventional space-filling (CPK) model.

184 Bond surfaces (intermediate density) may be applied to elucidate bonding and not only to portray known bonding. For example, the bond surface for diborane clearly shows a molecule with very little electron density concentrated between the two borons. This suggests that the appropriate Lewis structure is the one which lacks a boron-boron bond, rather than the one which has the two borons directly bonded.

185 Another important application of bond surfaces is to the description of the bonding in transition states. An example is the pyrolysis of ethyl formate, leading to formic acid and ethylene. + The bond surface offers clear evidence of a late transition state. The CO bond is nearly fully cleaved and the migrating hydrogen is more tightly bound to oxygen (as in the product) than to carbon (as in the reactant).

186 Spin density For open-shell molecules, the spin density indicates the distribution of unpaired electrons. Spin density is an obvious indicator of reactivity of radicals (in which there is a single unpaired electron). Bonds will be made to centers for which the spin density is greatest. For example, the spin density isosurface for allyl radical suggests that reaction will occur on one of the terminal carbons and not on the central carbon.

187 Electrostatic potential The value of the electrostatic potential (the energy of interaction of a positive point charge with the nuclei and electrons of a molecule) mapped onto an electron density isosurface may be employed to distinguish regions on the surface which are electron rich ( basic or subject to electrophilic attack) from those which are electron poor ( acidic or subject to nucleophilic attack). Negative potential surfaces serve to outline the location of the highest-energy electrons, for example lone pairs.

188 Example: A surface for which the electrostatic potential is negative, above and below the plane of the ring in benzene, and in the ring plane above the nitrogen in pyridine benzene pyridine While these two molecules are structurally very similar, potential surfaces make clear that this similarity does not carry over into their electrophilic reactivities.

189 Polarization potential The polarization potential provides the energy due to electronic reorganization of the molecule as a result of its interaction with a point positive charge. For example, It properly orders the proton affinities (measure of gas-phase basicity, or energy released when molecule accept a proton) of trimethylamine, dimethyl ether and fluoromethane. Local Ionization potential The local ionization potential is intended to reflect the relative ease of electron removal ( ionization ) at any location around a molecule. For example, a surface of low local ionization potential for sulfur tetrafluoride demarks the areas which are most easily ionized.

190 Molecular dynamics Molecular dynamics (MD) is computer simulation technique where the time evolution of atoms is followed by solving their equations of motions. It uses a Maxwell-Botzmann averaging for thermodynamic properties Results emerge in a form of simulation. Changes in structures of systems, vibrations, as well as movements of particles are simulated. Simulation brings to life the models yielding vast array of chemical and physical information often surpassing (in content at least) the real experiments.

191 Molecular dynamics basics The laws of classical mechanics are followed, most notably Newton s law: F i = m i a i for each atom i in a system constituted by N atoms. Here, m i is the atom mass, a i = acceleration, and F i the force acting upon it. d 2 r i dt 2 its MD is a deterministic technique: given an initial set of positions and velocities, the subsequent time evolution is in principle completely determined. In practice small numerical errors cause chaotic behaviour (butterfly effect). MD is a statistical mechanics method. It is a way to obtain a set of configurations or states distributed according to some statistical distribution function, or statistical ensemble. The properties, such as kinetic energy for example, are calculated using time averages. These are assumed to correspond to observable ensemble averages when the system is allowed to evolve in time indefinitely so system will eventually pass through all possible states (Ergodic hypothesis). Because the simulations are of fixed duration, one must be certain to sample a sufficient amount of phase space.

192 Phase Space For a system of N particles (e.g. atoms), the phase space is the 6N dimensional space of all the positions and momenta. At any given time, the state of the system (i.e. generally, the position and velocity of every atom) is given by a unique point in the phase space. The time evolution of the system can be seen as a displacement of the point in the phase space. Molecular dynamics as a simulation method is mainly a way of exploring, or sampling, the phase space. One of the biggest problem in molecular simulations is that the volume of the phase space (i.e., the number of accessible configurations for the system) is usually so huge that it is impossible to examine all of it. However, (for the case of a constant temperature system, usual in MD) different regions of the phase space have different probabilities to be observed. Boltzmann distribution says that the system has a higher probability to be in a low energy state. Molecular dynamics can be viewed as a way of producing configurations of the system (so, points in the phase space) according to their Boltzmann weight.

193 Modeling the system Choosing the potential energy function V(r 1,, r N ) Deriving the forces as the gradients of the potential with respect to atomic displacements: F i = r i V(r 1,, r N ) Writing the potential as a sum of pairwise interactions: V r 1,, r N = φ r i r j i j>i Introducing two-body interaction model, for example Lennard-Jones pair potential φ LJ r = 4ε σ r 12 σ r 6

194 In practice it is customary to establish a cutoff radius R c so that the potential becomes V r = φ LJ r φ LJ R c if r R c 0 if r > R c

195 Time integration The trajectories of interacting particles are calculated by integrating their equation of motion over time Time integration is based on finite difference methods, where time is discretized on a finite grid, the time step t being the distance between consecutive points in the grid. Knowing the positions and time derivatives at time t, the integration gives new quantities at a later time t + t. By iterating the procedure, the time evolution of the system can be followed for long times. The most commonly used time integration algorithm is propably the velocity Verlet algorithm, where position, velocities and accelerations at time t + t are obtained from the same quantities at time t in the following way r t + t = r t + v t t a t t 2 v t + t/2 = v t a t t a t + t = 1 m V(r(t + t)) v t + t = v t + t/ a t + t t Velocities are required (to obtain kinetic energy K) to test the conservation of energy E = K + V.

196

197 Application Areas for MD Materials Science Equilibrium thermodynamics Phase transitions Properties of lattice defects Nucleation and surface growth Heat/pressure processing Ion implantation Properties of nanostructures Medicine Drug design and discovery Chemistry Intra- and intermolecular interactions Chemical reactions Phase transitions Free energy calculations Biophysics and biochemistry Protein folding and structure prediction Biocombatibility (cell wall penetration, chemical processes) Docking

198 Different levels of methods QM-Based Methods o Limited to the range of hundreds of atoms o Very short times in dynamical simulations (<100ps) on a supercomputer. o Most reliable MD method applicable o Can be used for treatment of chemical reactions Classical MD o Whenever large systems, long time scales, and/or long series of events are needed for dynamical simulations, we must rely on classical (non-qm) MD Hybrid QM-MM o QM and classical MD can be applied simultaneously o System divided into parts

199 Examples: A proton transfer dynamics of 2-aminopyridine dimer studied using ab initio molecular dynamics (QM-MD) (Phys. Chem. Chem. Phys., 2011, 13, )

200 Examples: QM-MM MD was used to simulate microsolvation structure of Glycine in water Glycine

201 Limitations of molecular dynamics Electrons are not present explicitly in classical MD Realism of potential energy surfaces o Parameters are imperfect Classical description of atomic motions o Quantum effects can become significant in any system as soon as T is sufficiently low The limitations on the size (number of atoms) and time of the simulation constrain the range of problems that can be addressed by the MD method o The size of structural features of interest are limited to the size of the MD computational cell on the order of tens of nm. o Using modern computers it is possible to calculate timesteps. Therefore MD can only simulate processes that occur within ns. This is a serious limitation for many problems, for example thermally-activated processes. o Record: Largest MD simulation

202 Periodic boundary conditions Particles are enclosed in a box Box is replicated to infinity by rigid translation to the cartesian directions All these replicated particles move together but only one of them is represented in a computer program Each particle in the box is not only interacting with other particles in the same box, but also with their images in other boxes In practice the number of interactions is limited because cutoff radius of interaction potential Surface can be modeled by creating a slab, having periodic boundary conditions only in two directions