Computational Chemistry: Molecular Simulations with Chemical and Biological Applications. Prof. M. Meuwly, Dr. PA Cazade, Dr. M.-W.

Transcription

1 Computational Chemistry: Molecular Simulations with Chemical and Biological Applications Prof. M. Meuwly, Dr. PA Cazade, Dr. M.-W. Lee

2 Overview 1. Electronic Structure of Molecules 1.1 The Electronic Problem 1.2 The Hartree Fock Equations 1.3 Basis Sets 1.4 Solving the HF equations 1.5 Correlated Methods 1.6 Density Functional Theory 2. Molecular Simulations 2.1 Force Fields and Energy Functions 2.2 Molecular Dynamics Simulations (MD) 2.3 Analysis of MD Simulations 2.4 Monte Carlo Simulations (MC) 2.5 Free Energy Simulations

3 Overview Literature: Electronic Structure: Quantum Chemistry by Szabo/Ostlund (Dover) Quantum Chemistry by Ira Levine (Pearson) Molecular Simulations: Molecular Modelling by A. Leach (Prentice Hall) Understanding Molecular Simulation by Frenkel and Smit (AP) Computer Simulation of Liquids by Allen and Tildesley (OUP) Credits: Exercises 10 minute presentation

4 Overview Exercises: Throughout the semester see semester plan Applied examples with Gaussian09 and CHARMM Hours: Mon Thu 15-17

5 Overview How well do we need to describe intermolecular interactions in order to contribute to interpretation, understanding and prediction of chemical processes? Depending on the observable in question, what level of detail is required? What can we learn about intermolecular interactions from comparing simulation results with experiments?

6 Small Molecules Example: Bent versus Linear Methylene 1959 Herzberg and Shoosmith (Nature, 1959, Exp) conclude it is linear 1960 Foster and Boys (J. Chem. Phys., 1960, Comp) predict an angle of 128 o 1970 Bender and Schaefer III (J. Am. Chem. Soc., 1970, Comp) confirm bent structure (135 o ) 1971 Herzberg and Johns (J. Chem. Phys., 1971) reinterpret spectra and confirm bent structure. Currently accepted value is o (K Kuchitsu (ed) "Structure of Free Polyatomic Molecules - Basic Data" Springer, Berlin, 1998) Such structure determination relies nowadays on fitting spectroscopic data to a (model) Hamiltonian. Large Molecules 1960 Perutz and coworkers Structure of haemoglobin 3-dimensional Fourier synthesis at 5.5-A resolution, obtained by X-ray analysis Nature, 1960, 416 (1960) 1994 Schlichting and coworkers: Crystal Structure of Photolyzed Carbonmonoxy-Myoglobin Nature, 808 (1994) 2003 Anfinrud and coworkers: Watching a protein as it functions with 150-ps time-resolved X-ray crystallography Science, 300 (2003)

7 Introduction to Electronic Structure Calculations The Hartree Fock Equations

8 Molecular properties Transition States Reaction coords. Ab initio electronic structure theory Hartree-Fock (HF) Electron Correlation (MP2, CI, CC, etc.) Spectroscopic observables Geometry prediction Benchmarks for parametrization Assist Experimentalists Goal: Insight into chemical phenomena.

9 Computational Chemistry Rationalizing a strained allyl structure Usually cis but trans because of strain through binding to Pd

10 Computational Chemistry organic splitting of H 2

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19 The Problem What is a molecule? A molecule is composed of atoms, or, more generally a collection of charged particles, positive nuclei and negative electrons. The interaction between charged particles is described by; V ij = V (r ij ) = q i q j 4πε 0 r ij = q i q j r ij r ij q j Coulomb Potential q i Coulomb interaction between these charged particles is the only important physical force necessary to describe chemical phenomena.

20 But, electrons and nuclei are in constant motion In Classical Mechanics, the dynamics of a system (i.e. how the system evolves in time) is described by Newton s 2nd Law: F = ma dv dr = m d 2 r dt 2 F = force a = acceleration r = position vector m = particle mass In Quantum Mechanics, particle behavior is described in terms of a wavefunction, Ψ. Time-dependent Schrödinger Equation H ˆ Ψ = i Ψ t ˆ H ( i = 1; = h 2π) Hamiltonian Operator

21 Time-Independent Schrödinger Equation ˆ H (r,t) = ˆ H (r) Ψ(r,t) = Ψ(r)e iet / If H is time-independent, the timedependence of Y may be separated out as a simple phase factor. ˆ H (r)ψ(r) = EΨ(r) Time-Independent Schrödinger Equation Describes the stationary properties of electrons.

22 Hamiltonian for a system with N-particles H ˆ = T ˆ + V ˆ Sum of kinetic (T) and potential (V) energy T ˆ = N i=1 T ˆ i = N i=1 2 2m i i 2 = 2 i = 2 2 x i y i z i N i= m i x i y + 2 i z i 2 Kinetic energy Laplacian operator ˆ V = N N V ij = i=1 j>1 N N i=1 j>1 q i q j r ij Potential energy When these expressions are used in the time-independent Schrodinger Equation, the dynamics of all electrons and nuclei in a molecule or atom are taken into account.

23 Born-Oppenheimer Approximation So far, the Hamiltonian contains the following terms: + H ˆ = T ˆ n + T ˆ e + ˆ ˆ T n ˆ T e ˆ V ne ˆ V ee ˆ V nn V ne + ˆ V ee + ˆ V nn Since nuclei are much heavier than electrons, their velocities are much smaller. To a good approximation, the Schrödinger equation can be separated into two parts: One part describes the electronic wavefunction for a fixed nuclear geometry. The second describes the nuclear wavefunction, where the electronic energy plays the role of a potential energy.

24 Born-Oppenheimer Approx. cont. In other words, the kinetic energy of the nuclei can be treated separately. This is the Born-Oppenheimer approximation. As a result, the electronic wavefunction depends only on the positions of the nuclei. Physically, this implies that the nuclei move on a potential energy surface (PES), which are solutions to the electronic Schrödinger equation. Under the BO approx., the PES is independent of the nuclear masses; that is, it is the same for isotopic molecules. E 0 H. + H. H H Solution of the nuclear wavefunction leads to physically meaningful quantities such as molecular vibrations and rotations.

25 Limitations of the Born-Oppenheimer approximation The total wavefunction is limited to one electronic surface, i.e. a particular electronic state. The BO approx. is usually very good, but breaks down when two (or more) electronic states are close in energy at particular nuclear geometries. In such situations, a non-adiabatic wavefunction - a product of nuclear and electronic wavefunctions - must be used. In writing the Hamiltonian as a sum of electron kinetic and potential energy terms, relativistic effects have been ignored. These are normally negligible for lighter elements (Z<36), but not for the 4 th period or higher. By neglecting relativistic effects, electron spin must be introduced in an ad hoc fashion. Spin-dependent terms, e.g., spin-orbit or spin-spin coupling may be calculated as corrections after the electronic Schrödinger equation has been solved. The electronic Hamiltonian becomes, H ˆ = T ˆ e + V ˆ ne+ + V ˆ ee + V ˆ nn B.O. approx.; fixed nuclear coord.

26 Self-consistent Field (SCF) Theory GOAL: Solve the electronic Schrödinger equation, H e Ψ=EΨ. PROBLEM: Exact solutions can only be found for one-electron systems, e.g., H 2+. SOLUTION: Use the variational principle to generate approximate solutions. Variational principle - If an approximate wavefunction is used in H e Ψ=EΨ, then the energy must be greater than or equal to the exact energy. The equality holds when Ψ is the exact wavefunction. In practice: Generate the best trial function that has a number of adjustable parameters. The energy is minimized as a function of these parameters.

27 SCF cont. The energy is calculated as an expectation value of the Hamiltonian operator: ˆ E = Ψ H e Ψdτ Ψ Ψdτ Introduce bra-ket notation, Ψ H ˆ e Ψdτ = Ψ H ˆ e Ψ bra n Ψ Ψdτ = Ψ Ψ ket m right Combined bracket denotes integration over all coordinates. E = Ψ ˆ H e Ψ Ψ Ψ complex conjugate, left If the wavefunctions are orthogonal and normalized (orthonormal), Then, Ψ i Ψ j = δ ij δ ij = 1 δ ij = 0 E = Ψ ˆ H e Ψ (Kroenecker delta)

28 SCF cont. Antisymmetric wavefunctions can be written as Slater determinants. Since electrons are fermions, S=1/2, the total electronic wavefunction must be antisymmetric (change sign) with respect to the interchange of any two electron coordinates. (Pauli principle - no two electrons can have the same set of quantum numbers.) Consider a two electron system, e.g. He or H 2. A suitable antisymmetric wavefunction to describe the ground state is: Φ( 1,2) = φ 1 α(1)φ 2 β(2) φ 1 α(2)φ 2 β(1) Each electron resides in a spin-orbital, a product of spatial and spin functions. (Spin functions are orthonormal: α α = β β =1; α β = β α = 0) Interchange the coordinates of the two electrons, ( ) = φ 1 α(2)φ 2 β(1) φ 1 α(1)φ 2 β(2) ( ) = Φ( 1,2 ) Φ 2,1 Φ 2,1 (He: φ 1 =φ 2 = 1s) (H 2 : φ 1 = φ 2 = φ bonding MO )

29 SCF cont. A more general way to represent antisymmetric electronic wavefunctions is in the form of a determinant. For the two-electron case, ( ) = φ 1α(1) φ 2 β(1) Φ 1,2 φ 1 α(2) φ 2 β(2) = φ 1α(1)φ 2 β(2) φ 1 α(2)φ 2 β(1) For an N-electron N-spinorbital wavefunction, Φ SD = ( ) φ 2 (1) φ N (1) ( ) φ 2 (2) φ N (2) φ 1 1 φ 1 2 ( ) φ 2 (N) φ N (N) φ 1 N, φ i φ j = δ ij A Slater Determinant (SD) satisfies the antisymmetry requirement. Columns are one-electron wavefunctions, molecular orbitals. Rows contain the electron coordinates. One more simplification: The trial wavefunction will consist of a single SD (single configuration as opposed to multi configuration) Hartree-Fock equations derived from variational principle

30 Hartree-Fock Equations h ˆ i = i ˆ g ij = ˆ H e = 1 r i r j N h ˆ + i i =1 a N Z a R a r i N i j >i ˆ g ij + ˆ V nn One-electron operator: electron i, moving in the field of the nuclei. Two-electron operator: Electron-electron repulsion. Total Hamiltonian Calculation of the energy. E e = Φ ˆ H e Φ E e = A ˆ Π H ˆ e A ˆ Π = N 1 p= 0 Examine specific integrals: Φ ˆ V nn Φ = V nn ( 1) p Π ˆ H e ˆ P Π Expectation value over Slater Determinant Nuclear repulsion does not depend on electron coordinates.

31 For coordinate 1, Π h ˆ 1 Π = [ φ 1 (1)φ 2 (2) φ N (N)] h ˆ 1 φ 1 (1)φ 2 (2) φ N (N) Π ˆ g 12 Π = [ ] = φ 1 (1) ˆ h 1 φ 1 (1) φ 2 (2) φ 2 (2) φ N (N ) φ N (N ) = h 1 The one-electron operator acts only on electron 1 and yields an energy, h 1, that depends only on the kinetic energy and attraction to all nuclei. [ φ 1 (1)φ 2 (2) φ N (N)] g ˆ 12 [ φ 1 (1)φ 2 (2) φ N (N )] = φ 1 (1)φ 2 (2) ˆ g 12 φ 1 (1)φ 2 (2) φ 3 (3) φ 3 (3) φ N (N ) φ N (N ) = φ 1 (1)φ 2 (2) ˆ g 12 φ 1 (1)φ 2 (2) = J 12 Coulomb integral, J 12 : represents the classical repulsion between two charge distributions φ 12 (1) and φ 22 (2). Π g ˆ 12 P ˆ 12 Π = [ φ 1 (1)φ 2 (2) φ N (N)] g ˆ 12 [ φ 2 (1)φ 1 (2) φ N (N)] = φ 1 (1)φ 2 (2) ˆ g 12 φ 2 (1)φ 1 (2) φ 3 (3) φ 3 (3) φ N (N ) φ N (N ) = φ 1 (1)φ 2 (2) ˆ g 12 φ 2 (1)φ 1 (2) = K 12 Exchange integral, K 12 : no classical analogue. Responsible for chemical bonds.

32 The expression for the energy can now be written as: E e = N h i + 1 i =1 2 N N i j (J ij K ij ) + V nn Sum of one-electron, Coulomb, and exchange integrals, and V nn. To apply the variational principle, the Coulomb and Exchange integrals are written as operators, E e = N φ i h ˆ φ + 1 i i 2 i =1 N N i j ˆ J i φ j (2) = φ i (1) ˆ g 12 φ i (1) φ j (2) ( φ j J ˆ φ φ i j j K ˆ i φ ) j + V nn ˆ K i φ j (2) = φ i (1) ˆ g 12 φ j (1) φ i (2) The objective now is to find the best orbitals (φ i, MOs) that minimize the energy (or at least remain stationary with respect to further changes in φ i ), while maintaining orthonormality of φ i.

33 Employ the method of Langrange Multipliers: f (x 1,x 2, x N ) g(x 1,x 2, x N ) = 0 L(x 1,x 2, x N,λ) = f (x 1, x 2, x N ) λg(x 1,x 2, x N ) Optimize L such that L = 0, x i N ( ) L = E λ ij φ i φ j δ ij ij N ( ) L λ i = 0 δl = δe λ ij δφ i φ j + φ i δφ j = 0 ij Function to optimize. In terms of molecular orbitals, the Langrange function is: i =1 Rewrite in terms of another function. Define Lagrange function. Constrained optimization of L. Change in L with respect to small changes in φ i should be zero. Change in the energy with respect changes in φ i. N δe = ( δφ i ˆ + φ i i i ˆ ) N + ( δφ i i i J ˆ j ˆ + φ i ˆ K ˆ j j δφ i ) ij Constraint is orthogonality of the φ i

34 Define the Fock Operator, F i F ˆ i = h ˆ i + N j ( J ˆ j ˆ ) K j Effective one-electron operator, associated with the variation in the energy. δe = N i =1 ( δφ i F ˆ φ + φ i i i F ˆ i δφ ) i Change in energy in terms of the Fock operator. δl = N δφ i F ˆ i φ i + φ i F ˆ ( i δφ ) i λ ( ij δφ i φ j + φ i δφ ) j = 0 i=1 N ij According to the variational principle, the best orbitals, φ i, will make δl=0. After some algebra, the final expression becomes: ˆ F i φ i = N j λ ij φ j Hartree-Fock Equations

35 After a unitary transformation, λ ij 0 and λ ii ε i. ˆ F i φ i '= ε i φ i ' HF equations in terms of Canonical MOs and diagonal Lagrange multipliers. ε i = φ i ' ˆ F i φ i ' Lagrange multipliers can be interpreted as MO energies. Note: 1. The HF equations cast in this way, form a set of pseudo-eigenvalue equations. 2. A specific Fock orbital can only be determined once all the other occupied orbitals are known. 3. The HF equations are solved iteratively. Guess, calculate the energy, improve the guess, recalculate, etc. 4. A set of orbitals that is a solution to the HF equations are called Self-consistent Field (SCF) orbitals. 5. The Canonical MOs are a convenient set of functions to use in the variational procedure, but they are not unique from the standpoint of calculating the energy.

36 Basis Set Approximation In most molecular calculations, the unknown MOs are expressed in terms of a known set of functions - a basis set. Two criteria for selecting basis functions. I) They should be physically meaningful. ii) computation of the integrals should be tractable. It is common practice to use a linear expansion of Gaussian functions in the MO basis because they are easy to handle computationally. Each MO is expanded in a set of basis functions centered at the nuclei and are commonly called Atomic Orbitals. (Molecular orbital = Linear Combination of Atomic Orbitals - LCAO).

37 MO Expansion φ i = M α c αi χ α LCAO - MO representation Coefficients are variational parameters ˆ F i M c αi χ α = ε i c αi χ α α FC = SCε F αβ = χ α M α S αβ = χ α χ β ˆ F χ β HF equations in the AO basis Matrix representation of HF eqns. Roothaan-Hall equations (closed shell) F αβ - element of the Fock matrix S αβ - overlap of two AOs Roothaan-Hall equations generate M molecular orbitals from M basis functions. N-occupied MOs M-N virtual or unoccupied MOs (no physical interpretation)

38 Total Energy in MO basis E = N φ i h ˆ φ + 1 i i 2 i =1 N N i j ( φ i φ j g ˆ φ φ φ i j i φ j g ˆ φ j φ ) i +V nn Total Energy in AO basis E = N M c αi c βi χ α h ˆ χ + 1 i β 2 i =1 αβ αβγδ One-electron integrals, M 2 Two-electron integrals, M 4 Products of AO coeff form Density Matrix, D occ.mo N M D γδ = c γj c δj ; D αβ = c αi c βi j ij c αi c γj c βi c δj ( χ α χ γ g ˆ χ χ χ β δ α χ γ g ˆ χ δ χ β )+ V nn occ.mo i Computed at the start; do not change

39 General SCF Procedure Obtain initial guess for coeff., c αi,form the initial D γδ Form the Fock matrix Two-electron integrals Iterate Diagonalize the Fock Matrix Form new Density Matrix

40 Computational Effort Formally, the SCF procedure scales as M 4 (the number of basis functions to the 4th power). Accuracy As the number of functions increases, the accuracy of the Molecular Orbitals improves. As M, the complete basis set limit is reached Hartree-Fock limit. Result: The best single determinant wavefunction that can be obtained. (This is not the exact solution to the Schrodinger equation.) Practical Limitation In practice, a finite basis set is used; the HF limit is never reached. The term Hartree-Fock is often used to describe SCF calculations with incomplete basis sets.

41 Restricted and Unrestricted Hartree-Fock Restricted Hartree-Fock (RHF) For even electron, closed-shell singlet states, electrons in a given MO with α and β spin are constrained to have the same spatial dependence. Restricted Open-shell Hartree-Fock (ROHF) The spatial part of the doubly occupied orbitals are restricted to be the same. Unrestricted Hartree-fock (UHF) α and β spinorbitals have different spatial parts. Energy α β } Spinorbitals φ i σ(n) RHF Singlet ROHF Doublet UHF Doublet

42 Comparison of RHF and UHF R(O)HF α and β spins have same spatial part Wavefunction, Φ, is an eigenfunction of S 2 operator. For open-shell systems, the unpaired electron (α) interacts differently with α and β spins. The optimum spatial orbitals are different. Restricted formalism is not suitable for spin dependent properties. Starting point for more advanced calculations that include electron correlation. Can not describe dissociation appropriately UHF α and β spins have different spatial parts Wavefunction is not an eigenfunction of S 2. Φ may be contaminated with states of higher multiplicity (2S+1). E UHF E R(O)HF Yields qualitatively correct spin densities. Starting point for more advanced calculations that include electron correlation. Correct behaviour at long range.

43 Ab Initio (latin, from the beginning ) Quantum Chemistry Summary of approximations Born-Oppenheimer Approx. Non-relativistic Hamiltonian Use of trial functions, MOs, in the variational procedure Single Slater determinant Basis set, LCAO-MO approx. RHF, ROHF, UHF Consequence of using a single Slater determinant and the Self-consistent Field equations: Electron-electron repulsion is included as an average effect. The electron repulsion felt by one electron is an average potential field of all the others, assuming that their spatial distribution is represented by orbitals. This is sometimes referred to as the Mean Field Approximation. Electron correlation has been neglected!!!

44 Essential points HF is first order approximation (no correlation) Introduction of a basis set to represent MOs allows practical calculations In practice, convergence of observables with basis set size needs to be established. RHF not suitable for dissociation problems UHF suitable but beware of spin contamination (in particular for metal containing systems) So far single-reference calculations (only one Slater determinant). Probably appropriate for most organic molecules but not necessarily for metalcontaining ones.

45 Introduction to Electronic Structure Calculations Basis Sets

46 Basis Set Approximation MOs are expanded in terms of Atomic Orbitals φ i = M α c αi χ α LCAO - MO representation Coefficients are variational parameters φ i (MO) is initially unknown; describing (expanding) the MO as a combination of known (χ) AO functions. As M, reach the complete basis set limit; not an approximation. When M is finite, the representation is approximate. Two criteria for selecting basis functions. i) They should be physically meaningful. ii) computation of the integrals should be tractable.

47 χ ζ,n,l,m Slater Type Orbitals (STO) ( r,θ,ϕ) = NY ( l,m θ,ϕ)r n 1 e ζ r STO depends on quantum numbers n,l,m and zeta, ζ. Y l,m θ,ϕ ( ) Spherical harmonics; N - normalization Advantages: 1. Physically, the exponential dependence on distance from the nucleus is very close to the exact hydrogenic orbitals. 2. Ensures fairly rapid convergence with increasing number of functions. Disadvantages: 1. Three and four center integrals cannot be performed analytically. 2. No radial nodes. These can be introduced by making linear combinations of STOs. Practical Use: 1. Calculations of very high accuracy, atomic and diatomic systems. 2. Semi-empirical methods where 3- and 4-center integrals are neglected.

48 Gaussian Type Orbitals (GTO) χ ζ,n,l,m ( r,θ,ϕ) = NY ( l,m θ,ϕ)r 2n 2 l e ζ r 2 Polar coordinates GTO depends on quantum numbers n,l,m and exponent zeta, ζ. d-function has five components (Y 2,2,Y 2,1,Y 2,0,Y 2,-1,Y 2,-2 ). χ ζ,l x,l y,l z ( x,y,z)= Nx l x y l y z l z r 2n 2 l e ζ r 2 Cartesian coordinates In Cartesian coords., the angular dependence of the GTO is computed from the sum of l x, l y, and l z (l x +l y +l z =1, a p-orbital). d-function has six components (x 2, y 2, z 2, xy, xz, yz) in cartesian coord. These may be transformed to spherical functions plus one extra s-type function: (x 2 +y 2 +z 2 ). f-orbitals have 10 components, which may be transformed to the 7- pure spherical ones plus 3 p-type functions.

49 GTOs are inferior to STOs in three ways: 1. At the nucleus, the GTO has zero slope; the STO has a cusp. Behavior near the nucleus is poorly represented. 2. GTOs diminish too rapidly with distance. The tail behavior is poorly represented. 3. Extra d-, f-, g-, etc. functions (from Cart. rep.)may lead to linear dependence of the basis set. They are usually dropped when large basis sets are used. Advantage: GTOs have analytical solutions. Use a linear combination of GTOs to overcome these deficiencies.

50 Classification Minimum basis: Only enough functions are used to contain the the electrons of the neutral atoms (usually core plus valence orbitals). 1 st row: 1s, 2s, 2p 5-AOs 2 nd row: 1s, 2s, 3s, 2p, 3p 9-AOs Double Zeta (DZ) basis: Double the number of all basis functions. Hydrogen has two 1s-functions: 1s and 1s Li-Ne: 1s and 1s, 2s and 2s, 2p and 2p 2-AOs 10-AOs Think of 1s and 1s as inner and outer functions. The inner function has larger ζ exponent and is tighter, outer 1s has a smaller ζ, more diffuse.

51 DZ basis yields a better description of the charge distribution compared to a minimal basis. Consider HCN, σ π Charge distributions are different in different parts of the molecule. C-H σ-bond consists of the H 1s orbital and the C 2p z. CN π-bond is made up of C and N 2p x (and 2p y ) AOs. Because the π-bond is more diffuse, the optimal exponent ζ for p x (p y ) should be smaller than that for the more localized p z orbital. DZ basis has the flexibility (while the minimal basis does not) to describe the charge distribution in both parts of the molecule. The optimized AO coefficient (in MO expansion) of the tighter inner p z function on carbon will be larger in the C-H bond. The more diffuse outer p x and p y functions will have larger AO coefficients in the π-bond.

52 Split Valence Basis Sets Doubling the number of functions provides a much better description of bonding in the valence region. Doubling the number of functions in the core region improves the description of energetically important but chemically uninteresting core electrons. Split valence basis sets improve the flexibility of the valence region and use a single (contracted) set of functions for the core. VDZ double zeta 2x number of basis functions in valence region VTZ triplet zeta 3x VQZ quadruple zeta 4x V5Z quintuple zeta 5x V6Z sextuple zeta 6x

53 Polarization Functions Consider HCN, σ π H-C σ-bond: Electron distribution along the CH bond is different from the perpendicular direction. The H 1s orbital does not describe this behavior well. If p-functions are added to hydrogen, then the p z AO can improve the description of the CH bond. H 1s H 2p z p-functions induce a polarization of s-orbitals. d-function induce polarization of p-orbitals, etc. For a single determinantal wavefunction, the 1st set of polarization functions is by far the most important and will describe most if not all of the important charge polarization effects.

54 Polarization Functions cont. To describe charge polarization effects at the SCF level, add P-functions to H (one set) D-functions to Li-Ne, Na-Ar (one set 1 st row, 1-2 sets for 2 nd row) To recover a larger fraction of the dynamical correlation energy, multiple functions of higher angular momentum (d, f, g, h, i ) are essential. Electron correlation - energy is lowered by electrons avoiding each other. Two types: 1) Radial correlation - two electrons, one close to the nucleus the other farther away. Need basis functions of the same type but different exponent. (tight and diffuse p-functions, for example) 2) Angular correlation - Two electrons on opposite sides of the nucleus. Basis set needs functions with the same exponent but different angular momentum. For s-functions, need p-functions (and d, f, g..) to account for angular correlation. Radial Angular in importance.

55 Diffuse Functions Diffuse functions, s-, p-, and d-functions with small exponents are usually added for specific purposes. (1) Calculations on anions. (2) Dipole moment (3) Polarizability

56 Contracted Basis Sets Energy optimized basis sets have a disadvantage. Many functions go toward representing the energetically important but chemical uninteresting core electrons. Suppose 10s functions have been optimized for carbon. Start with 10 primitive gaussians PGTOs End with 3 contracted gaussians CGTOs Inner 6 describe core 1s electrons Next 4 describe valence electrons Contract to one 1s function contract to two 2s functions χ(cgto) = k i a i χ i (PGTO) Energy always increases! Fewer variational parameters. But, less CPU time required.

57 Pople Style Basis Sets STO-nG Minimal basis, n=# of gaussian primitives contracted to one STO. k-nlmg Split valence basis sets** 3-21G Contraction scheme (6s3p/3s) -> [3s2p/2s] (1 st row elements /H) 3 PGTOs contracted to 1, forms core 2PGTOs contracted to 1, forms inner valence 1 PGTO, forms outer valence After contraction of the PGTOs, C has 3s and 2p AOs. 6-31G (10s4p/4s) -> [3s2p/2s] Valence double zeta basis 6-311G (11s5p/4s) -> [4s3p/3s] Valence triple zeta basis 6-31+G* Equivalent to 6-31+G(d). 6-31G basis augmented with diffuse sp-functions on heavy atoms, polarization function (d) on heavy atoms G(2df,2pd) Triplet split valence; augmented with diffuse sp- on heavy atoms and diffuse s- on H s. Polarization functions 2d and 1f on heavy atoms; 2p and 1d on H s. (**In the Pople scheme, s- and p-functions have the exponent. 6-31G(d,p) most common)

58 Introduction to Electronic Structure Calculations Electron Correlation

59 What is electron correlation and why do we need it? Φ 0 is a single determinantal wavefunction. Φ SD = ( ) φ 2 (1) φ N (1) ( ) φ 2 (2) φ N (2) φ 1 1 φ 1 2 φ 1 N ( ) φ 2 (N) φ N (N) Slater Determinant, φ i φ j = δ ij Recall that the SCF procedure accounts for electron-electron repulsion by optimizing the one-electron MOs in the presence of an average field of the other electrons. The result is that electrons in the same spatial MO are too close together; their motion is actually correlated (as one moves, the other responds). E el.cor. = E exact - E HF (B.O. approx; non-relativistic H)

60 RHF dissociation problem Consider H 2 in a minimal basis composed of one atomic 1s orbital on each atom. Two AOs (χ) leads to two MOs (φ) H H φ 2 = N 2 (χ A χ B ); antibonding MO H 1s H 1s φ 1 = N 1 (χ A + χ B ); bonding MO H H 1 and 2 label the electrons; A and B the nuclei

61 The ground state wavefunction is: Φ 0 = φ 1α(1) φ 1 α(2) φ 1 β(1) φ 1 β(2) Slater determinant with two electrons in the bonding MO Φ 0 = φ 1 α(1)φ 1 β(2) φ 1 α(2)φ 1 β(1) [ ] Φ 0 = φ 1 (1)φ 1 (2) α(1)β(2) β(1)α(2) Φ 0 = φ 1 (1)φ 1 (2) = (χ A (1) + χ B (1))(χ A (2) + χ B (2)) Φ 0 = χ A (1)χ A (2) + χ B (1)χ B (2) + χ A (1)χ B (2) + χ B (1)χ A (2) Expand the Slater Determinant Factor the spatial and spin parts Only consider spatial part Four terms in the AO basis χ A χ A χ B χ B χ A χ B χ B χ A Ionic terms, two electrons in one Atomic Orbital, i.e. on one H center; H - Covalent terms, two electrons shared between two AOs

62 H 2 Potential Energy Surface E 0 H H Bond stretching H. + H. At the dissociation limit, H 2 must separate into two neutral atoms. H H At the RHF level, the wavefunction, Φ, is 50% ionic and 50% covalent at all bond lengths. χ A χ A χ A χ B χ B χ B χ B χ A H 2 does not dissociate correctly at the RHF level!! Should be 100% covalent at large internuclear separations.

63 RHF dissociation problem has several consequences: Energies for stretched bonds are too large. Affects transition state structures - E a are overestimated. Equilibrium bond lengths are too short at the RHF level. (Potential well is too steep.) HF method overbinds the molecule. Curvature of the PES near equilibrium is too great, vibrational frequencies are too high. The wavefunction contains too much ionic character; dipole moments (and also atomic charges) at the RHF level are too large. On the bright side, SCF procedures recover ~99% of the total electronic energy. But, even for small molecules such as H 2, the remaining fraction of the energy - the correlation energy - is ~110 kj/mol, on the order of a chemical bond.

64 To overcome the RHF dissociation problem, Use a trial function that is a combination of Φ 0 and Φ 1 First, write a new wavefunction using the anti-bonding MO. φ 2 = N 2 (χ A χ B ); antibonding MO The form is similar to Φ 0, but describes an excited state: Φ 1 = φ 2α(1) φ 2 β(1) φ 2 α(2) φ 2 β(2) = φ 2α(1)φ 2 β(2) φ 2 α(2)φ 2 β(1) [ ] Φ 1 = φ 2 (1)φ 2 (2) α(1)β(2) β(1)α(2) MO basis Φ 1 = φ 2 (1)φ 2 (2) = (χ A (1) χ B (1))(χ A (2) χ B (2)) Φ 1 = χ A (1)χ A (2) + χ B (1)χ B (2) χ A (1)χ B (2) χ B (1)χ A (2) AO basis Ionic terms Covalent terms

65 Trial function - Linear combination of Φ 0 and Φ 1 ; two electron configurations. Ψ = a 0 Φ 0 + a 1 Φ 1 = a 0 (φ 1 φ 1 ) + a 1 (φ 2 φ 2 ) Ψ = (a 0 + a 1 )[ χ A χ A + χ B χ B ]+ (a 0 a 1 )[ χ A χ B + χ B χ ] A Ionic terms Covalent terms Three points: 1. As the bond is displaced from equilibrium, the coefficients (a 0, a 1 ) vary until at large separations, a 1 = -a 0 : Ionic terms disappear and the molecule dissociates correctly into two neutral atoms. Ψ = Ψ CI, an example of configuration interaction. 2. The inclusion of anti-bonding character in the wavefunction allows the electrons to be farther apart on average. Electronic motion is correlated. 3. The electronic energy will be lower (two variational parameters).

66 Configuration Interaction - Excited Slater Determinants Since the HF method yields the best single determinant wavefunction and provides about 99% of the total electronic energy, it is commonly used as the reference on which subsequent improvements are based. As a starting point, consider as a trial function a linear combination of Slater determinants: Ψ = a 0 Φ HF + a i Φ i i =1 Multi-determinant wavefunction a 0 is usually close to 1 (~0.9). M basis functions yield M molecular orbitals. For N electrons, N/2 orbitals are occupied in the RHF wavefunction. M-N/2 are unoccupied or virtual (anti-bonding) orbitals.

67 Generate excited Slater determinants by promoting up to N electrons from the N/2 occupied to M-N/2 virtuals: a,b,c = virtual MOs b b a a a a,b c b a c,d 5 4 i i i,j k i k,l i i,j,k = occupied MOs 3 2 j j j 1 Excitation level Ψ HF Ref. Ψ i a Ψ ij ab Ψ ij ab abc Ψ ijk abcd Ψ ijkl Single Double Triple Quadruple

68 Represent the space containing all N-fold excitations by Ψ(N). Then the COMPLETE CI wavefunction has the form Where Ψ CI = C 0 Φ HF + Φ (1) + Φ (2) + Φ (3) Φ (N ) Φ HF = Hartree Fock Φ (1) = Φ (2) = Φ (3) = Φ (N ) = occ virt i occ a virt i, j occ i, j,k occ i, j,k... a,b virt a,b,c C i a Ψ i a C ij ab Ψ ij ab virt C abc abc ijk Ψ ijk a,b,c... C abc... abc... ijk... Ψ ijk... Linear combination of Slater determinants with single excitations Doubly excitations Triples N-fold excitation The complete Ψ CI expanded in an infinite basis yields the exact solution to the Schrödinger eqn. (Non-relativistic, Born-Oppenheimer approx.)

69 abc... C ijk... The various coefficients,, may be obtained in a variety of ways. A straightforward method is to use the Variation Principle. E CI = Ψ CI H Ψ CI Ψ CI Ψ CI Expectation value of H e. E CI abc... = 0 Cijk... Energy is minimized wrt coeff H C K = E KC K abc... The elements of the vector, C K, are the coefficients, C ijk... And the eigenvalue, E K, approximates the energy of the K th state. In a fashion analogous to the HF eqns, the CI Schrodinger equation can be formulated as a matrix eigenvalue problem. E 1 = E CI for the lowest state of a given symmetry and spin. E 2 = 1 st excited state of the same symmetry and spin, and so on.

70 Some nomenclature One-electron basis (one-particle basis) refers to the basis set. This limits the description of the one-electron functions, the Molecular Orbitals. The size of the many-electron basis (N-particle basis) refers to the number of Slater determinants. This limits the description of electron correlation. In practice, Complete CI (Full CI) is rarely done even for finite basis sets - too expensive. Computation scales factorially with the number of basis functions (M!). Full CI within a given one-particle basis is the benchmark for that basis since 100% of the correlation energy is recovered. Used to calibrate approximate correlation methods. CI expansion is truncated at a some excitation level, usually Singles and Doubles (CISD). Ψ CI = C 0 Φ HF + Φ (1) + Φ (2)

71 Number of configurations Example H 2 O: (19 basis functions) CISD (~80-90%) Full CI

72 Example: Neon Atom Relative importance Ref. Singles 2 Doubles 1 Triples 4 Quadruples 3 Weight = abc... (C abc... ijk... ) 2 ijk... for a given excitation level. (Frozen core approx., 5s4p3d basis - 32 functions) 1. CISD (singles and doubles) is the only generally applicable method. For modest sized molecules and basis sets, ~80-90% of the correlation energy is recovered. 2. CISD recovers less and less correlation energy as the size of the molecule increases.

73 Multi-configuration Self-consistent Field (MCSCF) H 2 O MOs Ψ HF Carry out Full CI and orbital optimization within a small active space. Six-electron in six-orbital MCSCF is shown. Written as [6,6]CASSCF. Complete Active Space Self-consistent Field (CASSCF) Why? 1. To have a better description of the ground or excited state. Some molecules are not welldescribed by a single Slater determinant, e.g. O To describe bond breaking/formation; Transition States. 3. Open-shell system, especially low-spin. 4. Low lying energy level(s); mixing with the ground state produces a better description of the electronic state. 5.

74 MCSCF Features: 1. In general, the goal is to provide a better description of the main features of the electronic structure before attempting to recover most of the correlation energy. 2. Some correlation energy (static correlation energy) is recovered. (So called dynamic correlation energy is obtained through CI and other methods through a large N-particle basis.) 3. The choice of active space - occupied and virtual orbitals - is not always obvious. (Chemical intuition and experience help.) Convergence may be poor. 4. CASSCF wavefunctions serve as excellent reference state(s) to recover a larger fraction of the dynamical correlation energy. A CISD calculation from a [n,m]-casscf reference is termed Multi-Reference CISD (MR- CISD). With a suitable active space, MRCISD approaches Full CI in accuracy for a given basis even though it is not size-extensive or consistent.

75 Examples of compounds that require MCSCF for a qualitatively correct description. H H C C H H Singlet state of twisted ethene, biradical. O + O O - zwitterionic O O O biradical H C N H C N H C N Transition State

76 Mœller-Plesset Perturbation Theory In perturbation theory, the solution to one problem is expressed in terms of another one solved previously. The perturbation should be small in some sense relative to the known problem. H ˆ = H ˆ 0 + λh ˆ ' H ˆ 0 Φ i = E i Φ i, i = 0,1,2,..., ˆ H Ψ = WΨ W = λ 0 W 0 + λ 1 W 1 + λ 2 W Ψ = λ 0 Ψ 0 + λ 1 Ψ 1 + λ 2 Ψ Hamiltonian with pert., λ Unperturbed Hamiltonian As the perturbation is turned on, W (the energy) and Ψ change. Use a Taylor series expansion in λ.

77 Define ˆ H 0 and ˆ H ' ˆ H 0 = ˆ H '= N N F ˆ i = h ˆ i + i =1 i=1 N i =1 N g ij j >1 N ( J ˆ ij K ˆ ) ij j =1 g ij N N i =1 j =1 W 0 = sum over MO energies W 1 = Φ 0 ˆ H ' Φ 0 W 2 = occ vir i < j a< b E(MP2) = = E(HF) Φ 0 H ˆ ab ' Φ ij occ vir i< j a <b E 0 E ij ab Φ ab ij H ˆ ' Φ 0 [ φ i φ j φ a φ b φ i φ j φ b φ ] 2 a ε i + ε j ε a ε b Unperturbed H is the sum over Fock operators Moller-Plesset (MP) pert th. Perturbation is a two-electron operator when H 0 is the Fock operator. With the choice of H 0, the first contribution to the correlation energy comes from double excitations. Explicit formula for 2nd order Moller-Plesset perturbation theory, MP2.

78 Advantages of MP n Pert. Th. MP2 computations on moderate sized systems (~150 basis functions) require the same effort as HF. Scales as M 5, but in practice much less. Size-extensive (but not variational). Size-extensivity is important; there is no error bound for energy differences. In other words, the error remains relatively constant for different systems. Recovers ~80-90% of the correlation energy. Can be extended to 4 th order: MP4(SDQ) and MP4(SDTQ). MP4(SDTQ) recovers ~95-98% of the correlation energy, but scales as M 7. Because the computational effort is significanly less than CISD and the size-extensivity, MP2 is a good method for including electron correlation.

79 Coupled Cluster Theory Perturbation methods add all types of corrections, e.g., S,D,T,Q,..to a given order (2nd, 3rd, 4th, ). Coupled cluster (CC) methods include all corrections of a given type to infinite order. The CC wavefunction takes on a different form: Ψ CC = e ˆ T Φ 0 Coupled Cluster Wavefunction Φ 0 is the HF solution T e ˆ = 1 ˆ + T ˆ ˆ T ˆ T 3 + = T ˆ = T ˆ + ˆ 1 T 2 + T ˆ ˆ T N 1 k! ˆ k =0 T k Exponential operator generates excited Slater determinants Cluster Operator N is the number of electrons

80 Comparison of Models CI-SD CI-SDTQ MP2 MP4(SDTQ) CCSD CCSD(T) Scale with M M 6 M 10 M 5 M 7 M 6 M 7 Size-extensive/consistent No ~Yes Y Y Y Y Variational Y Y No No No No Generally applicable Y No Y Y Y Y Requires ÔgoodÕzero-order Φ Y ~No Y Y ~No No Extension to Multi-reference Yes Yes Not yet common Accuracy with a medium sized basis set (single determinant reference): HF << MP2 < CISD < MP4(SDQ) ~CCSD < MP4(SDTQ) < CCSD(T) In cases where there is (a) strong multi-reference character and (b) for excited states, MR-CI methods may be the best option.

81 Effective Core Potential (ECP) Two good reasons to use ECP: (1) A balanced basis requires a proper description of the core and valence regions. Metals have a large number of electrons. Most of the computational effort is used to describe the energy but not the valence region. (2) For large Z, relativistic effects complicate matters. Solve both by using an ECP: Core electrons are modeled by a suitable potential function, and only the valence electrons are treated explicitly. LANL2DZ-ECP Stuttgart-ECP In the case of the Cs-ECPs, both also include the 5s and 5p filled shells explicity. The rest are considered core electrons.

82 Comparison between Expt. and Theory Three points: 1. There is excellent agreement between experiment and theory (CCSD(T) or MR-CISD) for of the E.A. for the atom and molecule Cs 2, the bond length and fundamental frequency of Cs Therefore, this basis set is likely to be suitable to describe the bonding in Cs 2 2- dimers and higher order clusters. 3. Justifies the use of ECPs since the comparison between expt and theory is quite good and what we are interested in is the charge distribution in the valence region.

83 Systematic Comparisons: H 2 O Geometry SCF MP2 CCSD(T) Expt. r(oh) = Å; θ = o

84 Convergence of Correlation Energy (H 2 O)

85 H 2 O Dipole Moment Diffuse functions essential

86 H 2 O Harmonic Frequencies HF ~ 6% too high MP2 Excellent agreement Expt cm -1, 3832 cm -1, 1649 cm -1 f -function improves bending freq. CCSD(T) Excellent agreement

87 Ozone, a problematic system Ozone Harmonic Frequencies Excellent agreement

88 Summary: The term chemical accuracy is used when a calculation has an error of ~1 kcal/mol. Chemical accuracy for almost any property of interest is best achieved with highly correlated wavefunctions (e.g. CCSD(T) or MR-CISD) and large basis sets (cc-pvtz and higher), and is only practical for small molecules. MP2 methods perform well for many properties of interest (geometry prediction, frequencies, dipole moment, ), is size extensive, recovers a good fraction of the correlation energy (80-90%), and is applicable to modest sized systems that contain 20 carbon atoms or more even with triple zeta basis sets,(cc-pvtz). MP2 does not perform well when the unperturbed state is multireference in nature (e.g. O 3 ). Multi-reference MP2 methods may be used in this case. Consult references at end for extended discussion.

89 References: Jensen, F.; Introduction to Computational Chemistry, John Wiley & Sons: Chichester, Another suitable reference for electronic structure theory: Szabo, A.; Ostlung, N.A. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill: New York, Quantum Chemistry texts: McQuarrie, D. A.; Simon, J. A. Physical Chemistry, A Molecular Approach; University Science Books: Sausalito, Lowe, J.P. Quantum Chemistry, Academic Press Inc.: New York, Basis Sets: Feller, D.; Davidson, E.R. Rev. Comput. Chem, 1990, 1. Davidson, E.R.; Feller, D. Chem. Rev. 1986, 86, Helgaker, T.; Taylor, P.R. Modern Electronic Structure Theory, Part II, ed. D.Yarkony, World Scientific: Gaussian Basis Sets on the web: