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Simplified Hartree-Fock Computations on Second-Row Atoms

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Simplified Hartree-Fock Computations on Second-Row Atoms Simplified Hartree-Fock Computations on Second-Row Atoms S. M. Blinder Wolfram Research, Inc. Champaign, IL 61820, USA May 18, 2021 Modern computational quantum chemistry has developed to a large extent beginning with applications of the Hartree-Fock method to atoms and molecules[1]. Density-functional methods have now largely supplanted conventional Hartree-Fock computations in current applications of computational chemistry. Still, from a pedagogical point of view, an un- derstanding of the original methods remains a necessary preliminary. However, since more elaborate Hartree-Fock computations are now no longer needed, it will suffice to consider a computationally simplified application of the method[2]. Accordingly, we will represent the many-electron wavefunction by a single closed-shell Slater determinant and use basis functions which are simple Slater-type orbitals. This is in contrast to the more elaborate Hartree-Fock computations, which might use multi-determinant wavefunctions and double- zeta basis functions. All of the symbolic and numerical operations were carried out using Mathematica. We consider Hartree-Fock computations on the ground states of the second-row atoms He through Ne, Z = 2 to 10, using the simplest set of orthonormalized 1s, 2s and 2p orbital functions: s α3=2 3β5 α + β = p e−αr; = 1 − r e−βr; 1s π 2s π(α2 − αβ + β2) 3 5=2 γ −γr 2pfx;y;zg = p re fsin θ cos φ, sin θ sin φ, cos θg: (1) arXiv:2105.07018v1 [quant-ph] 14 May 2021 π An orbital function multiplied by a spin function α or β is known as a spinorbital, a product of the form ( α φ(x) = (r) : (2) β 1 A simple representation of a many-electron atom is given by a Slater determinant con- structed from N occupied spin-orbitals: φa(1) φb(1) : : : φn(1) 1 φa(2) φb(2) : : : φn(2) Ψ(1;:::;N) = p ; (3) . .. N! . φa(N) φb(N) : : : φn(N) where the subscripts a; b; : : : stand for spinorbitals, for example, 1sα; 2pxβ, etc. and the arguments 1, 2, ::: are abbreviations for x1; x2;::: . The Hamiltonian for an N-electron atom, in hartree atomic units, is given by: N N X 1 Z X 1 H = − r2 − + : (4) 2 i r r i=1 i i>j ij The corresponding approximation to the total energy is then given by N N X X E = hΨjHjΨi = Hi + (Jij − Kij); (5) i=1 i>j where H, J and K are, respectively, the core, Coulomb and exchange integrals: Z 1 Z H = d3r ∗(r) − r2 − (r); (6) i i 2 r i ZZ j (r)j2j (r0)j2 J = d3rd3r0 i j ; (7) ij jr − r0j ZZ 1 K = d3r d3r0 ∗(r) ∗(r0) (r0) (r)hσ jσ i: (8) ij i j jr − r0j i j i j The exchange integral vanishes if σi 6= σj, in other words, if spinorbitals i and j have opposite spins, α; β or β; α. Explicit formulas for these integrals are given in the Appendix. Let us illustrate with a couple of examples. The core integral H2p is found from " # Z 1 Z π Z 2π γ5=2 1 Z γ5=2 p re−γr cos θ − r2 − p re−γr cos θ r2 sin θ dr dθ dφ, (9) 0 0 0 π 2 r π using the Laplacian in spherical polar coordinates 1 @ @ 1 @ @ 1 @2 r2 = r2 + sin θ + : (10) r2 @r @r r2 sin θ @θ @θ r2 sin2 θ @φ2 2 The symbolic computation can be carried out using Mathematica, giving the result 1 H = (γ2 − Zγ): (11) 2p 2 The Coulomb integral J1s;2p is found from Z 1 Z π Z 1 Z π α3=2 1 γ5=2 2 −αr1 −γr2 2 2 (2π) p e p r2e cos θ2 r1 sin θ1 dr1 dθ1 r2 sin θ2 dr2 dθ2 0 0 0 0 π r12 π (12) The interelectronic potential can be expanded in a series of Legendre polynomials: 1 l 1 X r< = l+1 Pl(cos Θ12); (13) r12 l=0 r> where r> and r< are the greater and lesser of r1 and r2. In the integral (12), the only nonva- r< nishing contribution is the l = 1 term 2 P1(cos Θ12), with P1(cos Θ12) = cos Θ12 = cos θ2. r> We must consider the contributions both with r2 > r1 and r1 > r2. Using Mathematica, the integral evaluates to 1 1 3α + γ J = − γ5 − + : (14) 1s;2p 2 γ4 (α + γ)5 The energy E is optimized by seeking a minimum as a function of the orbital parameters α, β and γ. For example, for Z = 6, the carbon atom with ground state configuration 2 2 0 3 1s 2s 2p 2p P0, we have E(α; β; γ) = 2H1s + 2H2s + 2H2p + J1s;1s + J2s;2s + 4J1s;2s − 2K1s;2s + 4J1s;2p − 2K1s;2p + 4J2s;2p − 2K2s;2p + J2p;2p0 − K2p;2p0 (15) Following is a tabulation of the results, showing the optimal parameters and the calcu- lated energies. Energies are expressed in atomic units. The computations made use of the FindMinimum function in Mathematica. Each computation took approximately 0.005 seconds of CPU time (using a MacBook Pro with the Apple M1 chip). For comparison, we also include the results of the best Hartree-Fock computations[3] and the exact nonrel- ativistic energies of the atomic ground states. Our approximate energies are within 1% of the accurate H-F values, obtained with an order of magnitude less computational effort. 3 Appendix: Core, Coulomb and Exchange Integrals 4 References [1] S. M. Blinder, \Introduction to the Hartree-Fock Method," in Mathematical Physics in Theoretical Chemistry (S. M. Blinder and J. E. House, eds.), Elsevier, 2018. [2] This paper is based in the Wolfram Demonstration: https://demonstrations.wolfram.com/SimplifiedHartreeFockComputationsOnSecondRowAtoms/ [3] Atomic Data and Nuclear Data Tables, 95(6), 2009 pp. 836-870. 5 Addendum: Approximation of Exchange-Correlation Energy Density-functional theory is based on the total electronic charge density ρ(r), which is a function of the coordinates of a single electron. By contrast the wavefunction for an N- electron system depends on N particle coordinates. The second Hohenberg-Kohn theorem states that there exists a functional of the density which gives a lower bound on the ground- state energy. The precise form of this functional is not known. Computations are based on constructed forms which give various levels of approximation of the exact energy. Kohn and Sham developed an alternative computational scheme which is a hybrid between DFT and the Hartree-Fock method. In particular, the complicated many-electron potential in the Hartree-Fock equation for a spinorbital is replaced by an effective potential VKS(r) incorporating Coulomb, exchange and correlation effects. The actual many-electron system is thereby represented by an equivalent system with independent particles, such that a single Slater determinant gives the exact solution. The Hartree-Fock-Kohn-Sham equation for a spinorbital φn(r) can be written n 1 o − r2 + V (r) φ (r) = φ (r): 2 KS n n n The Kohn-Sham equations do not contain any coupling terms involving other orbitals, in contrast to the exchange interactions in the Hartree-Fock equations. After some numerical experimentation, we propose here a very simple approximation to the Coulomb-exchange-correlation potential Z 2 0 2 X j nr)j j m(r )j V (r) = λ d3r0 ; KS jr − r0j m6=n with the parameter λ ≈ 0:94 Following is a tabulation of the computed results using the \λ approximation." The re- sulting ground-state energies are usually within 0.1% of the exact values. 6 Z best H-F λ approx exact 2 -2.86168 -2.879 -2.903385 3 -7.43273 -7.484 -7.477976 4 -14.5730 -14.67 -14.668449 5 -24.5291 -24.69 -24.658211 6 -37.6886 -37.89 -37.855668 7 -54.4009 -54.57 -54.611893 8 -74.8094 -75.05 -75.109991 9 -99.4093 -99.64 -99.803888 10 -128.547 -128.7 -128.830462 7.
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