E depends on the nuclear positions through the

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E depends on the nuclear positions through the nuclear-electron attraction and nuclear-nuclear repulsion terms E = 0 corresponds to all particles at infinite separation nuclei B A AB B A electrons j i ij nuclei A iA A electrons i i electrons i e el r Z Z e r e r Z e m 2 2 2 2 2 2 ˆ H d d E E el el el el el el el el * * ˆ , ˆ H H
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its.unc.edu 18 Approximate Wavefunctions Approximate Wavefunctions Construction of one-electron functions (molecular orbitals, MO’s) as linear combinations of one- electron atomic basis functions (AOs) MO-LCAO approach. Construction of N-electron wavefunction as linear combination of anti-symmetrized products of MOs (these anti-symmetrized products are denoted as Slater-determinants). down) - (spin up) - (spin ; 1 i i u i k N k kl i l r q
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its.unc.edu 19 The Slater Determinant The Slater Determinant z c b a z c b a z c b a z c b a z z z z c c c c b b b b a a a a n z c b a z c b a n z c b a n n n n n n n n 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 1 2 3 2 1 3 2 1 Α ˆ ! 1 ! 1
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its.unc.edu 20 The Two Extreme Cases The Two Extreme Cases One determinant: The Hartree–Fock method. All possible determinants: The full CI method. N N 3 2 1 3 2 1 HF There are N MOs and each MO is a linear combination of N AOs. Thus, there are n N coefficients u kl , which are determined by making stationary the functional: The ij are Lagrangian multipliers . N l k ij lj kl ki N j i ij u S u H E 1 , * 1 , HF HF HF ˆ
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its.unc.edu 21 The Full CI Method The Full CI Method The full configuration interaction (full CI) method expands the wavefunction in terms of all possible Slater determinants: There are possible ways to choose n molecular orbitals from a set of 2N AO basis functions. The number of determinants gets easily much too large. For example: n N 2 1 ˆ ; 2 1 , CI CI CI 2 1 CI  c S c H E c n N * n N 9 10 10 40 Davidson’s method can be used to find one or a few eigenvalues of a matrix of rank 10 9 .
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its.unc.edu 22 N N 3 2 1 3 2 1 HF N l k ij lj kl ki N j i ij u S u H E 1 , * 1 , HF HF HF ˆ N i li ki kl N l k kl mn N n m mn u u P nl mk P h P E H 1 * 1 , 2 1 1 , nuc HF HF ; ˆ ki 0 HF E u ki Hartree–Fock equations The Hartree–Fock Method The Hartree–Fock Method
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its.unc.edu 23  | S Overlap integral         | 2 1 | P H F  i i occ i c c 2 P Density Matrix   S F i i i c c The Hartree–Fock Method The Hartree–Fock Method
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its.unc.edu 24 1.
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