Computational Physics

4.6 Basis functions 63 of the orbital parts only, read J ̃(r)φ(r)= 2 ∑ l ∫ d^3 r′φ∗l(r′)φl(r′) 1 |r′−r| φ(r) and K ̃(r)φ(r)= ∑ l ...

64 The Hartree–Fock method It is convenient to introduce the density matrix which (for RHF) is defined as Ppq= 2 ∑ k CpkC∗qk. (4 ...

4.6 Basis functions 65 model the exact solutions to the Fock equations accurately. A molecule consists of atoms which, in isolat ...

66 The Hartree–Fock method Figure 4.2. Positions in the Gaussian product theorem(4.79). These functions are calledprimitive basi ...

4.6 Basis functions 67 are only five d-states! This paradox is solved by noticing that the linear combination (x^2 +y^2 +z^2 )e− ...

68 The Hartree–Fock method φ1s 0.25 0.5 0 0.5 1 1.5 2 r STO-2G STO-3G STO-4G STO Figure 4.3. Approximation of a 1s Slater orbita ...

4.7 The structure of a Hartree–Fock computer program 69 point for this calculation is the atomic orbitals of the ground state of ...

70 The Hartree–Fock method fact thatp,qcan be interchanged withr,smeans that the range ofscan be restricted tos≤q. All in all, a ...

4.7 The structure of a Hartree–Fock computer program 71 Diagonalise the Fock matrix; Construct a new density matrixPfrom the ei ...

72 The Hartree–Fock method Counting all different configurations of the indices, likep=r
=q,sandq
=s, and so on, 14 different ca ...

4.8 Integrals involving Gaussian functions 73 The convergence can finally be enhanced by extrapolating the values of the density ...

74 The Hartree–Fock method The kinetic integral:This is given by 〈1s,α,A|−∇^2 |1s,β,B〉= ∫ d^3 r∇e−α(r−RA) 2 ∇e−β(r−RB) 2 (4.101) ...

4.8 Integrals involving Gaussian functions 75 Furthermore, the Fourier transform of exp(−γr^2 )is(π/γ )^3 /^2 exp(−k^2 / 4 γ),so ...

76 The Hartree–Fock method The functionF 0 (t)can be evaluated using the error function erf, which is avail- able in most high-l ...

4.9 Applications and results 77 Table 4.1.Bond lengths in atomic units for three different molecules. Hartree–Fock (HF) and expe ...

78 The Hartree–Fock method Table 4.2.Bond angles in degrees forH 2 OandNH 3. The angles are those of theH–O–HandH–N–H chains res ...

4.10 Improving upon the Hartree–Fock approximation 79 Table 4.4.Ionisation potentials in atomic units for various molecules. Res ...

80 The Hartree–Fock method Table 4.5.Correlation energies in atomic units forH 2 andH 2 O. Molecule CI Exact H 2 −0.039 69 −0.04 ...

Exercises 81 freedom). For this function we make the followingAnsatz: (Rn,ri)=χ(Rn)(ri) with(ri)an eigenstate with eigenvalue ...

82 The Hartree–Fock method We construct the functionsandχfrom the orthonormal spatial orbitalsφ 1 (r), φ 2 (r)and the spin-up a ...