Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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The matrix elements, ˜H kk ′, of the Hamiltonian: ˜H kk ′ = 〈B k |H|B k ′〉, H = −¯h2 ∇ 2 1 2m e − Zq2 e − ¯h2 ∇ 2 2 − Zq2 e r 1 2m e r 2 + q2 e r 12 (D.27) can be written using matrix elements H ij = Hij K + HV ij , obtained for the hydrogen-like case with Z = 2, Eq.(5.26) and (5.27): ˜H kk ′ = ( H ii ′S jj ′ + H ij ′S ji ′ + S ii ′H jj ′ + H ij ′S ji ′) + 〈Bk |V ee |B k ′〉, (D.28) and the matrix element of the Coulomb electron-electron interaction V ee : 〈B k |V ee |B k ′〉 = ∫ ∫ + b i(k) (r 1 )b j(k) (r 2 ) q2 e b r i(k ′ )(r 1 )b j(k ′ )(r 2 )d 3 r 1 d 3 r 2 12 b i(k) (r 1 )b j(k) (r 2 ) q2 e r 12 b j(k ′ )(r 1 )b i(k ′ )(r 2 )d 3 r 1 d 3 r 2 . These matrix elements can be written, using Eq.(7.33), as (D.29) 〈B k |V ee |B k ′〉 = qeπ 2 5/2 αβ(α + β) 1/2 + qeπ 2 5/2 α ′ β ′ (α ′ + β ′ ) 1/2 , (D.30) where α = α i(k) + α i(k ′ ), β = α j(k) + α j(k ′ ), α ′ = α i(k) + α j(k ′ ), β ′ = α j(k) + α i(k ′ ). (D.31) In an analogous way one can calculate the matrix elements between symmetrized products of gaussians formed with P-type gaussian functions (those defined in Eq.5.23). The combination of P-type gaussians with L = 0 has the form: B k (r 1 , r 2 ) = √ 1 ( ) (r 1 · r 2 ) b i(k) (r 1 )b j(k) (r 2 ) + b i(k) (r 2 )b j(k) (r 1 ) 2 (D.32) It is immediately verified that the product of a S-type and a P-type gaussian yields an odd function that does not contribute to the ground state. In the case with S-type gaussians only, the code writes to file ”gs-wfc.out” the function: P (r 1 , r 2 ) = (4πr 1 r 2 ) 2 |ψ(r 1 , r 2 )| 2 , (D.33) where P (r 1 , r 2 )dr 1 dr 2 is the joint probability to find an electron between r 1 and r 1 + dr 1 , and an electron between r 2 and r 2 + dr 2 . The probability to find an electron between r and r + dr is given by p(r)dr, with ∫ ∫ p(r) = 4πr 2 |ψ(r, r 2 )| 2 4πr2dr 2 2 = P (r, r 2 )dr 2 . (D.34) It is easy to verify that for a wave function composed by a product of two identical functions, like the one in (D.10), the joint probability is the product of single-electron probabilities: P (r 1 , r 2 ) = p(r 1 )p(r 2 ). This is not true in general for the exact wave function. 94
D.3.1 Laboratory • observe the effect of the number of basis functions, to the choice of coefficients λ of the gaussians, to the inclusion of P-type gaussians • compare the obtained energy with the one obtained by other methods: perturbation theory with hydrogen-like wave functions, (Sec.D.1), variational theory with effective Z (Sec.D.3), exact result (-5.8074 Ry). • Make a plot of the probability P (r 1 , r 2 ) and of the difference P (r 1 , r 2 ) − p(r 1 )p(r 2 ), using for instance gnuplot and the following commands: set view 0, 90 unset surface set contour set cntrparam levels auto 10 splot [0:4][0:4] "gs-wfc.out" u 1:2:3 w l Note that the probability P (r 1 , r 2 ) (column 3 in ”splot”) is not exactly equal to the product p(r 1 )p(r 2 ) (column 4; column 5 is the difference between the two). 95
Page 1 and 2:
Lecture notes Numerical Methods in
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3.3 Code: crossection . . . . . . .
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Introduction The aim of these lectu
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is an important and well-known libr
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Chapter 1 One-dimensional Schrödin
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Under the assumption of Eq.(1.8), E
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point its value is still ∼ 60% of
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an equation involving finite differ
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eigenvalue) to force the code to pe
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1.3.3 Laboratory Here are a few hin
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The Hamiltonian can thus be written
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In the following we will use q 2 e
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The dominating term close to the nu
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whose goal is to give an estimate,
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Chapter 3 Scattering from a potenti
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3.2 Scattering of H atoms from rare
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1. Solution of the radial Schrödin
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• In the limit of a weak potentia
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By modifying ψ as ψ + δψ, the e
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This demonstrates Eq.(4.19), since
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that the same equations, for a fini
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eported here are immediately genera
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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 and 60: Now that we have the expectation va
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 the corresponding energy is E =
Page 101: • Relative error: |a − b| < ɛa
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Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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