Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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where we have introduced a ”Hartree potential” V H (it is not the same as in the Hartree equations!) and an ”exchange potential” V x . The Hartree potential is the same for all orbitals: V H (1) = ∑ j ∫ ∫ φ ∗ j(2) q2 e φ j (2)dv 2 ≡ r 12 where we have introduce the charge density ρ(2) q2 e r 12 dv 2 , (7.14) ρ(2) = ∑ j φ ∗ j(2)φ j (2). (7.15) We can verify that ρ(1) is equal to the probability to find an electron in (1), that is, ∫ ρ(1) = N |Ψ(1, 2, .., N)| 2 dv 2 ...dv N . (7.16) The exchange term: ( ˆV x φ k )(1) = − ∑ j ∫ δ(σ k , σ j ) φ ∗ j(2) q2 e r 12 φ k (2)φ j (1) dv 2 (7.17) does not have the simple form of the Hartree potential: V H (1)φ k (1), where V H (1) comes from an integration over variable 2. It has instead a form like ∫ ( ˆV x φ)(1) ≡ V x (1, 2)φ k (2)dv 2 (7.18) typical of a non local interaction. 7.1.2 Correlation energy The Hartree-Fock solution is not exact: it would be if the system under study were described by a wave function having the form of a Slater determinant. This is in general not true. The energy difference between the exact and Hartree- Fock solution is known as correlation energy. 1 The origin of the name comes from the fact that the Hartree-Fock approximation misses part of the ”electron correlation”: the effects of an electron on all others. This is present in Hartree- Fock via the exchange and electrostatic interactions; more subtle effects are not accounted for, because they require a more general form of the wave function. For instance, the probability P (r 1 , r 2 ) to find an electron at distance r 1 and one as distance r 2 from the origin is not simply equal to p(r 1 )p(r 2 ), because electrons try to ”avoid” each other. The correlation energy in the case of He atom is about 0.084 Ry: a small quantity relative to the energy (∼ 1.5%), but not negligible. An obvious way to improve upon the Hartree-Fock results consists in allowing contributions from other Slater determinants to the wave function. This is the essence of the “configuration interaction” (CI) method. Its practical application requires a sophisticated ”technology” to choose among the enormous 1 Feynman called it stupidity energy, because the only physical quantity that it measures is our inability to find the exact solution! 56
number of possible Slater determinants a subset of most significant ones. Such technique, computationally very heavy, is used in quantum chemistry to get high-precision results in small molecules. Other, less heavy methods (the socalled Møller-Plesset, MP, approaches) rely on perturbation theory to yield a rather good estimate of correlation energy. A completely different approach, which produces equations that are reminiscent of Hartree-Fock equations, is Density-Functional Theory (DFT), much used in condensed-matter physics. 7.1.3 The Helium atom The solution of Hartree-Fock equations in atoms also commonly uses the central field approximation. This allows to factorize Eqs.(7.11) into a radial and an angular part, and to classify the solution with the ”traditional” quantum numbers n, l, m. For He, the Hartree-Fock equations, Eq.(7.11), reduce to ∫ − ¯h2 ∇ 2 2m 1φ 1 (1) − Zq2 e φ 1 (1) + e r 1 ∫ + φ ∗ 1(2) q2 e r 12 [φ 1 (2)φ 1 (1) − φ 1 (2)φ 1 (1)] dv 2 φ ∗ 2(2) q2 e r 12 [φ 2 (2)φ 1 (1) − δ(σ 1 , σ 2 )φ 1 (2)φ 2 (1)] dv 2 = ɛ 1 φ 1 (1) (7.19) Since the integrand in the first integral is zero, ∫ − ¯h2 ∇ 2 2m 1φ 1 (1) − Zq2 e φ 1 (1) + φ ∗ e r 2(2) q2 e [φ 2 (2)φ 1 (1) 1 r 12 −δ(σ 1 , σ 2 )φ 1 (2)φ 2 (1)] dv 2 = ɛ 1 φ 1 (1). (7.20) In the ground state, the two electrons have opposite spin (δ(σ 1 , σ 2 ) = 0) and occupy the same spherically symmetric orbital (that is: φ 1 and φ 2 are the same function, φ). This means that the Hartree-Fock equation, Eq.(7.20), for the ground state is the same as the Hartree equation, Eq.(6.19). In fact, the two electrons have opposite spin and there is thus no exchange. In general, one speaks of Restricted Hartree-Fock (RHF) for the frequent case in which all orbitals are present in pairs, formed by a same function of r, multiplied by spin functions of opposite spin. In the following, this will always be the case. 7.2 Code: helium hf gauss The radial solution of Hartree-Fock equations is possible only for atoms or in some model systems. In most cases the solution is found by expanding on a suitable basis set, in analogy with the variational method. Let us re-write the Hartree-Fock equation – for the restricted case, i.e. no total spin – in the following form: Fφ k = ɛ k φ k , k = 1, . . . , N/2 (7.21) The index k labels the coordinate parts of the orbitals; for each k there is a spin-up and a spin-down orbital. F is called the Fock operator. It is of 57
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 and 58: Chapter 7 The Hartree-Fock approxim
Page 59: Now that we have the expectation va
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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