Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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where the “integral” on dv 1 means as usual “integration on coordinates, sum over spin components” . We follow the same path of Sec. (6.2) used to derive Hartree equations, Eq.(6.13). Since a determinant for N electrons has N! terms, we need a way to write matrix elements between determinants on a finite paper surface. The following property, valid for any (symmetric) operator F and determinantal functions ψ and ψ ′ , is very useful: 〈ψ|F |ψ ′ 〉 = 1 ∫ ∣ ∣∣∣∣∣ φ ∗ 1 (1) . φ∗ 1 (N) ∣ ∣∣∣∣∣∣ . . . F N! φ ∗ N (1) . φ∗ N (N) ∣ ∫ = φ ∗ 1(1) . . . φ ∗ N(N)F ∣ φ ′ 1 (1) . φ′ 1 (N) . . . φ ′ N (1) . φ′ N (N) ∣ ∣∣∣∣∣∣ dv 1 . . . dv N φ ′ 1 (1) . φ′ 1 (N) . . . φ ′ N (1) . φ′ N (N) ∣ ∣∣∣∣∣∣ dv 1 . . . dv N (7.4) (by expanding the first determinant, one gets N! terms that, once integrated, are identical). From the above property it is immediate (and boring) to obtain the matrix elements for one- and two-electron operators: 〈ψ| ∑ f i |ψ〉 = ∑ ∫ φ ∗ i (1)f 1 φ i (1) dv 1 (7.5) i i (as in the Hartree approximation), and 〈ψ| ∑ g ij |ψ〉 = ∑ ∫ φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − φ j (1)φ i (2)] dv 1 dv 2 (7.6) 〈ij〉〈ij〉 The integrals implicitly include summation over spin components. If we assume that g 12 depends only upon coordinates (as in Coulomb interaction) and not upon spin, the second term: ∫ φ ∗ i (1)φ ∗ j(2)g 12 φ j (1)φ i (2) dv 1 dv 2 (7.7) is zero is i and j states are different (the spin parts are not affected by g 12 and they are orthogonal if relative to different spins). It is convenient to move to a scheme in which the spin variables are not explicitly included in the orbital index. Eq.(7.6) can then be written as 〈ψ| ∑ g ij |ψ〉 = ∑ ∫ φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − δ(σ i , σ j )φ j (1)φ i (2)] dv 1 dv 2 〈ij〉〈ij〉 where σ i is the spin of electron i, and: (7.8) δ(σ i , σ j ) = 0 if σ i ≠ σ j = 1 if σ i = σ j In summary: 〈ψ|H|ψ〉 = ∑ ∫ i + ∑ ∫ 〈ij〉 φ ∗ i (1)f 1 φ i (1) dv 1 (7.9) φ ∗ i (1)φ ∗ j(2)g 12 [φ i (1)φ j (2) − δ(σ i , σ j )φ j (1)φ i (2)] dv 1 dv 2 54
Now that we have the expectation value of the energy, we can apply the variational principle. In principle we must impose normalization constraints such that not only all φ i stay normalized (as we did in the derivation of Hartree’s equation) but also all pairs φ i , φ j with same spin are orthogonal, i.e., a (triangular) matrix ɛ ij of Lagrange multipliers would be needed. It can be shown however (details e.g. on Slater’s book, Quantum theory of matter) that it is always possible to find a transformation to a solution in which the matrix of Lagrange multipliers is diagonal. We assume that we re dealing with such a case. Let us omit the details of the derivation and move to the final Hartree-Fock equations: f 1 φ k (1) + ∑ ∫ φ ∗ j(2)g 12 [φ k (1)φ j (2) − δ(σ k , σ j )φ j (1)φ k (2)] dv 2 = ɛ k φ k (1) j (7.10) or, in explicit form, − ¯h2 2m e ∇ 2 1φ k (1) − Zq2 e r 1 φ k (1) + ∑ j ∫ φ ∗ j(2) q2 e r 12 [φ j (2)φ k (1) (7.11) − δ(σ k , σ j )φ k (2)φ j (1)] dv 2 = ɛ k φ k (1) The energy of the system, Eq. 7.9, can be expressed, analogously to the Hartree case, via the sum of eigenvectors of Eq.(7.11), minus a term compensating the double counting of Coulomb repulsion and of the exchange energy: E = ∑ ɛ k − ∑ ∫ φ ∗ k(1)φ ∗ j(2)g 12 [φ k (1)φ j (2) − δ(σ j , σ k )φ j (1)φ k (2)] dv 1 dv 2 . k (7.12) Eq.(7.11) has normally an infinite number of solutions, of which only the lowestenergy N will be occupied by electrons, the rest playing the role of excited states. The sum over index j runs only on occupied states. Let us carefully observe the differences with respect to Hartree equations, Eqs.(6.13): 1. ∑ j also includes the case j = k. 2. for electrons j with same spin as that of k, there is an additional exchange term 3. due to the exchange term, the case j = k yields a nonzero contribution only if the spins are different. 7.1.1 Coulomb and exchange potentials Let us analyze the physical meaning of the Hartree-Fock equations. We re-write them under the form − ¯h2 2m e ∇ 2 1φ k (1) − Zq2 e r 1 φ k (1) + V H (1)φ k (1) + ( ˆV x φ k )(1) = ɛ k φ k (1), (7.13) 55
Page 1 and 2:
Lecture notes Numerical Methods in
Page 3 and 4:
3.3 Code: crossection . . . . . . .
Page 5 and 6:
Introduction The aim of these lectu
Page 7 and 8: is an important and well-known libr
Page 9 and 10: Chapter 1 One-dimensional Schrödin
Page 11 and 12: Under the assumption of Eq.(1.8), E
Page 13 and 14: point its value is still ∼ 60% of
Page 15 and 16: an equation involving finite differ
Page 17 and 18: eigenvalue) to force the code to pe
Page 19 and 20: 1.3.3 Laboratory Here are a few hin
Page 21 and 22: The Hamiltonian can thus be written
Page 23 and 24: In the following we will use q 2 e
Page 25 and 26: The dominating term close to the nu
Page 27 and 28: whose goal is to give an estimate,
Page 29 and 30: Chapter 3 Scattering from a potenti
Page 31 and 32: 3.2 Scattering of H atoms from rare
Page 33 and 34: 1. Solution of the radial Schrödin
Page 35 and 36: • In the limit of a weak potentia
Page 37 and 38: By modifying ψ as ψ + δψ, the e
Page 39 and 40: This demonstrates Eq.(4.19), since
Page 41 and 42: that the same equations, for a fini
Page 43 and 44: eported here are immediately genera
Page 45 and 46: BLAS 5 (collected here: dgemm.f 6 )
Page 47 and 48: known as generalized eigenvalue pro
Page 49 and 50: The Hamiltonian operator for this p
Page 51 and 52: Chapter 6 Self-consistent Field A w
Page 53 and 54: (the variations of all other normal
Page 55 and 56: nucleus. Even in open-shell atoms,
Page 57: Chapter 7 The Hartree-Fock approxim
Page 61 and 62: number of possible Slater determina
Page 63 and 64: Let us now introduce a further vari
Page 65 and 66: The index R is a reminder that both
Page 67 and 68: Molecular orbitals theory can expla
Page 69 and 70: Verify how sensitive the result is
Page 71 and 72: potential, we can hope to obtain a
Page 73 and 74: with wave vector k will be purely k
Page 75 and 76: Fast Fourier Transform In the simpl
Page 77 and 78: truncate it, it is convenient to co
Page 79 and 80: A reciprocal lattice of vectors G m
Page 81 and 82: The origin of the coordinate system
Page 83 and 84: Chapter 11 Exact diagonalization of
Page 85 and 86: The only nonzero matrix elements fo
Page 87 and 88: The code requires the number N of s
Page 89 and 90: Appendix A From two-body to one-bod
Page 91 and 92: Appendix B Accidental degeneracy an
Page 93 and 94: and [J 2 1 , J z] = [J 2 2 , J z] =
Page 95 and 96: part and symmetric coordinate part)
Page 97 and 98: and the corresponding energy is E =
Page 99 and 100: D.3.1 Laboratory • observe the ef
Page 101: • Relative error: |a − b| < ɛa
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Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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