Yingwei Wang Computational Quantum Chemistry 1 Hartree energy 2. 2 Many-body system 2. 3 Born-Oppenheimer approximation 2

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1 Purdue University CHM Computational Quantum Chemistry REVIEW Yingwei Wang October 10, 2013 Review: Prof Slipchenko s class, Fall 2013 Contents 1 Hartree energy 2 2 Many-body system 2 3 Born-Oppenheimer approximation 2 4 Variational principle 4 5 Orbitals Kinds of orbitals Notations of spin orbitals Hartree product Hartree-Fock energy Two electrons Many electrons Properties Self-consistent procedure 8 8 Basis sets Basic concepts Examples Koopmans theorem 11 1

2 1 Hartree energy The hartree (symbol: Eh or Ha), also known as the Hartree energy, is the atomic unit of energy, named after the British physicist Douglas Hartree. 2 E h = m e a 2, (1.1) 0 = J, (1.2) = ev, (1.3) = kcal/mol. (1.4) Table 1: Energy levels Phenomenon Energy level nuclear physics E h ionization, excitation, dissociation, chemistry reaction E h vibration E h rotation E h spin < 10 4 E h 2 Many-body system The many body Hamiltonian is Ĥ = ˆT e + ˆT N + ˆV ee + ˆV NN + ˆV en. (2.1) Table 2: Items in many body Hamiltonian Name Notation Sign i 2 i Kinetic energy of electron ˆTe = 1 2 Kinetic energy of nuclei ˆTN = 1 Electron repulsion ˆVee = i>j Nuclear repulsion ˆVNN = A>B 1 Electron-nuclear attraction ˆV en = A,i 2M A A 2 A Negative sign comes from momentum operator p = i 1 r ij Positive sign comes from charge ( e)( e) = e 2 r AB Z A r Ai Negative sign comes from charge Z A ( e)e 3 Born-Oppenheimer approximation Main idea The core of Born-Oppenheimer approximation is that electron and nuclear degrees of freedom are separable. Electronic Schrödinger Equation After Born-Oppenheimer approximation, the electronic Hamiltonian Ĥel is defined as follows: 2

3 Table 3: Changes in Born-Oppenheimer approximation Name Notation Changes Kinetic energy of electron ˆTe = 1 2 i 2 i unchanged Kinetic energy of nuclei ˆTN = 1 2M A A 2 A 0 since M A m e Electron repulsion ˆVee = 1 i>j r ij unchanged Nuclear repulsion ˆVNN = A>B 1 r AB becomes a constant Electron-nuclear attraction ˆVeN = Z A A becomes parametric dependence, A,i r Ai instead of variable dependence. Table 4: Operators about electronic Hamiltonian Operators Notations 0-electron operator V NN = Z AZ B A>B r AB 1-electron operator ĥ(i) = i Z A A r ia 2-electron operator ˆv(i,j) = 1 r ij Electronic Hamiltonian Ĥ el = iĥ(i)+ i>j 1 r ij +V NN Exception Born-Oppenheimer approximation fails when the electron and nuclei motion is coupled. An example is malondialdehyde CH 2 (CHO) 2, where H is not only a particle but also a wave. See Fig.1. Figure 1: Malondialdehyde CH 2 (CHO) 2 3

4 4 Variational principle The variational principle has two parts: 1. The complete set of exact eigenfunctions of a Hamiltonian H define an orthogonal complete basis set for the total space of wave functions. If so, it is possible to express any function as a linear combination of the exact eigenfunctions. 2. If φ is any normalised, well-behaved function that satisfies the boundary conditions of the Hamiltonian, then the energy associated with φ is always above that of the eigenfunction of lowest energy, i.e. φ H φ E 0, (4.1) where E 0 is the true value of the lowest energy eigenvalue of H. This principle allows us to calculate an upper bound for the ground state energy by finding the trial wavefunction φ for which the integral is minimised (hence the name trial wavefunctions are varied until the optimum solution is found). 5 Orbitals 5.1 Kinds of orbitals Table 5: Orbitals Name Notation Functions Orbital a wave function for a single particle, a electron Spatial orbital φ i (r) describes the spatial distribution of an electron such that φ i (r) 2 dr is the probability of finding the particle near r Spin orbital χ i (x) = φ(r)α(ω) or φ(r)β(ω) a wave function for an electron that describes both its distribution and spin Molecular orbitals the wave functions for electrons in molecular Hartree product ψ HP = χ i (x 1 )χ j (x 2 ) χ k (x p ) Note: spin orbitals, instead of spatial obitals 5.2 Notations of spin orbitals Sometimes it is convenient to use a notation that indicates a spin orbital by its spatial part, using a bar or lack of bar to denote whether it has the β or α spin. In this notation, the Hartree Fock ground state of H 2 is ψ 0 = χ 1 χ 2 = φ 1 φ1 = 1 1, which indicates that both electrons occupy the same spatial orbital φ 1, but one has α spin and the other has β spin. 5.3 Hartree product Proposition 5.1. Define the Hartree product Ψ HP (x 1,x 2,,x N ) = χ i (x 1 )χ j (x 2 ) χ k (x N ). (5.1) 4

5 Consider the Hamiltonian in which H HP = h(i), (5.2) i=1 h(i)χ j (x i ) = ε j χ j (x i ). (5.3) Shown that the Hartree product (5.1) is an eigenfunction of the Hamiltonian (5.2) with an eigenvalue given by E = ε i +ε j + +ε k. (5.4) Proof. According to (5.3) and (5.1), we have where E is given by (5.4). h(i)ψ HP = ε i Ψ HP, h(i)ψ HP = (ε i +ε j + +ε k )Ψ HP, i=1 H HP Ψ HP = EΨ HP, 6 Hartree-Fock energy 6.1 Two electrons The Hamiltonian of system with 2-electrons is ( Ĥel N=2 = ) ( Z A + 1 r 1A A A The trail function of slater type is The corresponding energy is Z A r 2A ) + 1 r 12 (6.1) = ĥ(1)+ĥ(2)+ 1 r 12. (6.2) ψ 0 = 1 2 (χ 1 (1)χ 2 (2) χ 1 (2)χ 2 (1)). (6.3) E N=2 HF = ψ 0 ĤN=2 el ψ 0, (6.4) = d1χ 1(1)ĥ(1)χ 1(1)+ d2χ 2(2)ĥ(2)χ 2(2) + d1d2χ 1 (1)χ 2 (2) 1 χ 1 (1)χ 2 (2) d1d2χ 1 r (1)χ 2 (2) 1 χ 2 (1)χ 1 (2)(6.5) 12 r 12 = h 1 +h 2 +J 12 K 12, (6.6) where J 12 is called coulomb term and K 12 is called exchange term. 5

6 6.2 Many electrons The energy of N electron system is E HF = ψ 0 Ĥel ψ 0 (6.7) = h i + (J ij K ij ), (6.8) = i=1 h i i=1 where J ij and K ij are defined in (6.10)-(6.13). 6.3 Properties (ij kl) = ij kl = Define some operators i=1 j>i i=1 j=1 (J ij K ij ), (6.9) dr 1 dr 2 ψ i(r 1 )ψ j (r 1 )r 1 12 ψ k(r 2 )ψ l (r 2 ), (6.10) dr 1 dr 2 ψ i (r 1)ψ j (r 2)r 1 12 ψ k(r 1 )ψ l (r 2 ), (6.11) J ij = (ii jj) = ij ij, called coulomb integral (6.12) K ij = (ij ji) = ij ji, called exchange integral. (6.13) Ĵ i x j (2) = x i (1) 1 x i (1) x j (2), r 12 (6.14) ˆK i x j (2) = x i (1) 1 x j (1) x i (2), r 12 (6.15) ˆf i = ĥi + (Ĵj ˆK j ), Fock operator, (6.16) x i ˆf i x i = ĥi + j (J ij K ij ). (6.17) Proposition 6.1. Prove the following properties of coulomb and exchange integrals: j J ii = K ii, (6.18) J ij = J ij, (6.19) K ij = K ij, (6.20) J ij = J ji, (6.21) K ij = K ji. (6.22) Proof. According to the notations (6.10)-(6.13), it is easy to know the following. J ii = (ii ii) = K ii ; J ij = (ii jj) = J ij ; 6

7 K ij = (ji ij), = dr 1 dr 2 ψj (r 1)ψ i (r 1 )r12 1 ψ i (r 2)ψ j (r 2 ), = dr 1 dr 2 ψj(r 2 )ψ i (r 2 )r12 1 ψ i(r 1 )ψ j (r 1 ), = dr 1 dr 2 ψi(r 1 )ψ j (r 1 )r12 1 ψ j(r 2 )ψ i (r 2 ), = (ij ji) = K ij ; J ij = (ii jj), = dr 1 dr 2 ψi (r 1)ψ i (r 1 )r12 1 ψ j (r 2)ψ j (r 2 ), = dr 1 dr 2 ψi (r 2)ψ i (r 2 )r12 1 ψ j (r 1)ψ j (r 1 ), = dr 1 dr 2 ψj(r 1 )ψ j (r 1 )r12 1 ψ i(r 2 )ψ i (r 2 ), = (jj ii) = J ji ; K ij = (ij ji), = dr 1 dr 2 ψi (r 1)ψ j (r 1 )r12 1 ψ j (r 2)ψ i (r 2 ), = dr 1 dr 2 ψi (r 2)ψ j (r 2 )r12 1 ψ j (r 1)ψ i (r 1 ), = dr 1 dr 2 ψj(r 1 )ψ i (r 1 )r12 1 ψ i(r 2 )ψ j (r 2 ), = (ji ij) = K ji. Proposition 6.2. Verify the energies of the determinants shown in Fig.2 by inspection. a. h 11 +h 22 +J 12 K 12, b. h 11 +h 22 +J 12, c. 2h 11 +J 11, d. 2h 2 +J 22, e. 2h 11 +h 22 +J 11 +2J 12 K 12, f. 2h 22 +h 11 +J 22 +2J 12 K 12, g. 2h 11 +2h 22 +J 11 +J 22 +4J 12 2K 12. Proof. Please see Table 6. Remark 6.1. Different spin means no exchange K ij. 7

8 Figure 2: Energy determinants Table 6: Interpretation of determinantal energies Number one-electron Coulomb Exchange a h 11 +h 22 J 12 K 12 b h 11 +h 22 J 12 0 c 2h 11 J 11 0 d 2h 2 J 22 0 e 2h 11 +h 22 J 11 +2J 12 K 12 f 2h 22 +h 11 J 22 +2J 12 K 12 g 2h 11 +2h 22 J 11 +J 22 +4J 12 2K 12 7 Self-consistent procedure Nonlinear Hartree-Fock equations can be solved by a self-consistent procedure, as follows 1. An initial guess orbitals (matrix C 0 ) are generated; F(C)C = SCE (7.1) 2. Fock operator is calculated for this set of orbitals (F 0 = F(C 0 )); 3. Fock operator is diagonalized, F 0 C 1 = SC 1 E 1 (called Roothaan equations), and new orbitals C 1 replace the orbitals C 0 from previous step; 4. Steps 2-3 are repeated until the difference between orbitals from step n and step n+1 is below certain (user-specified) threshold. Remark 7.1. A typical sailing of HF method is: diagonalization K 3 and form Fock matrix K 4. 8 Basis sets 8.1 Basic concepts Slater type and Gassuian type Double zeta and triple zeta 1. Polarization functions: adding d-type basis functions to the heavy atoms (, Li-F) and p-type functions to hydrogen ( ). Note that He does not need polarization. 2. Diffusion functions: (1) Charged particles, such as HSO 4 ; (2) He 2. Remark 8.1. Polarization is more expensive than diffusion. 8

9 8.2 Examples Question: Determine the total number of basis functions and primitive basis functions for STO-3G, 6-31G, 6-31G**, and 6-311(+,+)G** calculations of methane (CH 4 ). Write the contraction scheme in general notations for each basis (Szabo & Ostlund, chapter 3.6). Assume that pure angular momentum (5d,7f etc functions) polarization functions are used. Answer: 1. The STO-3G basis set for methane consists of one 1s orbital on each H atom and a 1s, 2s, and set of three 2p orbitals on C. See Tables 7-9 for details. The contraction scheme in general notations is (6s3p/3s)[2s1p/1s]. Table 7: STO-3G: Carbon atom 1s 2s 2p(3) Total No. Basis Functions No. Gaussians for each orbital No. Primitives Table 8: STO-3G: Hydrogen atom 1s Total No. Basis Functions 1 1 No. Gaussians for each orbital 3 No. Primitives 3 3 Table 9: STO-3G: Methane molecule No. Basis Functions = 9 No. Primitives = The 6-31G basis set is a split valence double zeta basis set. For hydrogen atom, a split valence double zeta basis consists of two 1s orbitals, denoted 1s and 1s. Note that for the H atom, because the 1s electron is considered the valence shell a double zeta basis set is used. For the carbon atom, a split valence double zeta basis set consists of a single 1s orbital, along with two 2s and two each of 2px, 2py, and 2pz orbitals, denoted as 1s, 2s, 2s, 2p(3) and 2p (3). See Tables for details. The contraction scheme in general notations is (10s4p/4s)[3s2p/2s]. Table 10: 6-31G: Carbon atom 1s 2s 2s 2p(3) 2p (3) Total No. Basis Functions No. Gaussians for each orbital No. Primitives

10 Table 11: 6-31G: Hydrogen atom 1s 1s Total No. Basis Functions No. Gaussians for each orbital 3 1 No. Primitives Table 12: 6-31G: Methane molecule No. Basis Functions = 17 No. Primitives = The 6-31G** basis set is a split valence double zeta basis set with added polarization functions on both carbon and hydrogen atoms. For hydrogen atom, the split valence double zeta basis consists of two 1s orbitals, and extra polarization functions 2px, 2py, and 2pz orbitals, denoted 1s, 1s and 2p(3). For the carbon atom, a split valence double zeta basis set consists of a single 1s orbital, along with two 2s and a double set of 2p orbitals, 2p(3)and 2p. (3), andextrapolarizationfunctions d(5). See Tables13-15fordetails. The contraction scheme in general notations is (10s4p1d/4s1p)[3s2p1d/2s1p]. Table 13: 6-31G**: Carbon atom 1s 2s 2s 2p(3) 2p (3) d(5) Total No. Basis Functions No. Gaussians for each orbital No. Primitives Table 14: 6-31G**: Hydrogen atom 1s 1s 2p(3) Total No. Basis Functions No. Gaussians for each orbital No. Primitives Table 15: 6-31G**: Methane molecule No. Basis Functions = 34 No. Primitives = 55 10

11 4. The 6-311(+,+)G** basis set is a split valence triple zeta basis set with added diffuse and polarization functions on both carbon and hydrogen atoms. For hydrogen atom, the split valence triple zeta basis consists of three 1s orbitals 1s,1s,1s, and diffuse function 2s, and extra polarization functions 2px, 2py, and 2pz orbitals, 2p(3). For the carbon atom, a split valence triple zeta basis set consists of a single 1s orbital, along with three 2s orbitals 2s,2s,2s, and a triple set of 2p orbitals, 2p(3), 2p (3) and 2p (3), and diffuse functions 3s, 3p(3), and extra polarization functions 3d(5). See Tables for details. The contraction scheme in general notations is (12s6p1d/6s1p)[5s4p1d/4s1p]. Table 16: 6-311(+,+)G**: Carbon atom 1s 2s 2s 2s 2p(3) 2p (3) 2p (3) 3s 3p d(5) Total No. Basis Functions No. Gaussians for each orbital No. Primitives Table 17: 6-311(+,+)G**: Hydrogen atom 1s 1s 1s 2s 2p(3) Total No. Basis Functions No. Gaussians for each orbital No. Primitives Table 18: 6-311(+,+)G**: Methane molecule No. Basis Functions = 50 No. Primitives = 71 9 Koopmans theorem Theorem 9.1 (Koopmans). This theorem has two parts: 1. The ionization potential (IP) for removing an electron from χ c is just the negative of the orbital energy ε c, IP = N 1 E c N E 0 = ε c > 0. (9.1) 2. The electron affinity (EA) for adding an electron to the virtual spin orbital χ r is just the negative of the orbital energy of that virtual spin orbital, i.e. EA = N E 0 N+1 E r = ε r. (9.2) If ε r is negative (i.e. if N+1 Ψ r ) is more stable than N Ψ 0, the electron affinity is positive. 11