Introduction to Computational Chemistry

Transcription

1 Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry room B430, Chemicum 4th floor September 3, 2013 Introduction and theoretical backround September 4 1 / 18

2 Overview of this course lectures include basic theory results of chemical problems student home work traditional home exercises write a report of some research topic oral presentation computer exercises final exam computational chemistry in supercomputer enviroment Introduction and theoretical backround September 4 2 / 18

3 What is computational chemistry? relies on results of theoretical chemistry and computer science practice of efficient computer programs to calculate numerical data of molecular systems and solids some of the current hot topics are ground level information to atmospheric phenomena (aerosol nucleation, atmospheric heat balance, reaction rates) nano chemistry (carbon nanotubes, semiconductors, metal clusters) drug design main 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? Introduction and theoretical backround September 4 3 / 18

4 About CSC CSC IT Center for Science Ltd is administered by the Ministry of Education, Science and Culture. CSC provides IT support and resources for academia, research institutions and companies. Researchers can use the largest collection of scientific software and databases in Finland. Introduction and theoretical backround September 4 4 / 18

5 Methods of computational chemistry Ab initio uses rigorous quantum mechanics + accurate and versatile computationally expensive good results for small systems Semi empirical uses approximate quantum mechanics relies on empirical or ab initio parameters big systems with hundreds of atoms Molecular mechanics Uses classical mechanics relies on empirical force fields no electronic properties and thus no bond breaking or forming large systems with thousands of atoms Introduction and theoretical backround September 4 5 / 18

6 Scrödinger equation HΨ = EΨ. (1) 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 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 Introduction and theoretical backround September 4 6 / 18

7 Variational method Yields approximate solution for the Scrödinger 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 E ground Φ H Φ (2) Proof: Trial function φ expanded as a linear combination of the exact eigenfunctions Ψ. Φ = c k Ψ k (3) The energy corresponding to this trialfunction is k E[Φ] = Φ H Φ Φ Φ (4) Introduction and theoretical backround September 4 7 / 18

8 Variational method Substituting the expansion over the exact wavefuntions, i,j E[Φ] = c ic j Ψ i H Ψ j c = c i,j ic j E j Ψ i Ψ j i,j ic j Ψ i Ψ j c i,j ic j Ψ i Ψ j = ic ic i E i i c ic i (5) We now subtract the exact ground state energy E 0 from both sides to obtain i E[Φ] E 0 = c ic i (E i E 0 ) i c (6) ic i E[Φ] E 0 (7) Any variations in the trial function which lower its energy are making the approximate energy closer to the exact answer. The solution can be obtained by optimizing (i.e. searching for the minimum) the parameters c i E/ c i = 0 (8) Introduction and theoretical backround September 4 8 / 18

9 Variational method Example: For a helium atom we can choose the trial function as follows: parameters: p, q, α Φ(r 1,r 2 ) = C[1+pr 12 +q(r 1 r 2 ) 2 ]exp[ α(r 1 +r 2 )] (9) electron coordinates: r 1,r 2,r 12 r 1 r 12 p = 0.30,q = 0.13,α = r 2 After optimization: E = (three parameters) E = (1024 parameters) E = (Experimental value) Introduction and theoretical backround September 4 9 / 18

10 Variational method In reality the trial function written as a linear combination of some basis functions which are not eigenfunctions Φ = c i φ i (10) i The variational parameters are the expansion coefficients c i. The energy for this approximate wavefunction is i,j E[Φ] = c ic j φ i H φ j c (11) i,j ic j φ i φ j which can be simplified using the notation H i,j = φ i H φ j (12) to yield E[Φ] = S i,j = φ i φ j (13) i,j c ic j H i,j i,j c ic j S i,j (14) Introduction and theoretical backround September 4 10 / 18

11 Variational method In order to the find the minimum value of E we differentiate with respect to the expansion coefficients c i and set values to 0 in each case. E = c jh j i,j c i c c js j i,j E i,j ic j S i,j c = 0, (15) i,j ic j S i,j If an orthonormal basis is used, the above equation is greatly simplified because S ij is 1 for i = j and 0 for i j. In this case, we can write it as an secular determinant as H 11 E H 12 H 1N H 21 H 22 E H 2N... H N1 H N2 H NN E = 0. (16) 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. Introduction and theoretical backround September 4 11 / 18

12 Born-Oppenheimer approximation Hamilton operator includes the kinetic and potential energy parts of the electrons and nuclei. Ĥ = ˆT + ˆV (17) Kinetic energy operator for N particles (electrons and nuclei) is ˆT = h2 2 N i 2 i m i, (18) where 2 i = 2 x 2 i + 2 y 2 i + 2 z 2 i and m i is the mass if i:th particle. Potential energy part includes Coulombic nucleus electron interaction, electron electron interaction, and nucleus nucleus interaction. Because electron is lighter than the proton by the factor 2000, the electron quickly rearranges in response to the slower motion of the nuclei. The total wave function for the molecule can be factored into its electronic and nuclear components. Introduction and theoretical backround September 4 12 / 18

13 Born-Oppenheimer approximation Example for H + 2 -ion, potential energy operator can be written as ˆV = e2 4πε 0 r 1 e2 4πε 0 r 2 + e 2 4πε 0 R AB, (19) where R AB is the distance between two nuclei, r 1 is the distance between the nucleus A and electron, r 2 is the distance between the nucleus B. The Scrödinger equation for the H + 2 -ion can be written according to the BO-approximation in simple form Ĥ (el) = h2 2 e 2m e 2 j=1 e 2 4πε 0 r j, (20) where first term is electronic kinetic energy operator. This equation can be further simplified by using dimensionless variables (atomic units) by setting electron mass and charge to one. The atomic unit of energy is called hartree. Ĥ (el) = 2 e 2 2 j=1 1 r j. (21) Introduction and theoretical backround September 4 13 / 18

14 Born-Oppenheimer approximation Three important equations Introduction and theoretical backround September 4 14 / 18

15 Born-Oppenheimer approximation 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) Introduction and theoretical backround September 4 15 / 18

16 Born-Oppenheimer approximation B-O potential energy surface of molecular electronic excited state Conical intersection B-O potential energy surface of molecular electronic ground state Molecule dissociates Introduction and theoretical backround September 4 16 / 18

17 Electron orbitals Electrons are assumed to locate in spin-orbitals which are one-particle wavefunctions φ(r,σ) = φ(ξ) taking both the position r and spin angular momentum σ (α =+ 1 or β = 2 1 ) as its coordinates. 2 The Pauli exclusion principle states: Electrons cannot occupy the same quantum states. Example: Wavefunction for two electrons (1 and 2) Ψ = 1 2 [φ 1 (1)φ 2 (2) φ 2 (1)φ 1 (2)]. (22) The heart of Pauli exclusion principle: Valid electronic wavefunctions must change sign upon exchanging the coordinates of any two electrons. Equation (22) can be written as determinant Ψ = 1 2 φ 1 (1) φ 2 (1) φ 1 (2) φ 2 (2). (23) Introduction and theoretical backround September 4 17 / 18

18 Electron orbitals In general, N el -electron molecular orbital is expressed as Slater determinant: φ 1 (1) φ 2 (1) φ N (1) Ψ = 1 φ 1 (2) φ 2 (2) φ N (2) Nel!.., (24). φ 1 (N) φ 2 (N) φ N (N) Consequence of the Pauli principle Without the Pauli principle matter would collapse and occupy a much smaller space. Fortunately, electrons of the same spin are kept apart by a repulsive short-range force. This exchange interaction which is additional to the long-range electrostatic force is responsible for the everyday observation that two objects cannot be in the same place in the same time. Introduction and theoretical backround September 4 18 / 18