Restricted Hartree-Fock Energy Research Articles

Arbitrarily accurate quantum alchemy.

Doping compounds can be considered a perturbation to the nuclear charges in a molecular Hamiltonian. Expansions of this perturbation in a Taylor series, i.e., quantum alchemy, have been used in the literature to assess millions of derivative compounds at once rather than enumerating them in costly quantum chemistry calculations. So far, it was unclear whether this series even converges for small molecules, whether it can be used for geometry relaxation, and how strong this perturbation may be to still obtain convergent numbers. This work provides numerical evidence that this expansion converges and recovers the self-consistent energy of Hartree-Fock calculations. The convergence radius of this expansion is quantified for dimer examples and systematically evaluated for different basis sets, allowing for estimates of the chemical space that can be covered by perturbing one reference calculation alone. Besides electronic energy, convergence is shown for density matrix elements, molecular orbital energies, and density profiles, even for large changes in electronic structure, e.g., transforming He3 into H6. Subsequently, mixed alchemical and spatial derivatives are used to relax H2 from the electronic structure of He alone, highlighting a path to spatially relaxed quantum alchemy. Finally, the underlying code that allows for arbitrarily accurate evaluation of restricted Hartree-Fock energies and arbitrary order derivatives is made available to support future method development.

Correlation energy of two electrons in a ball

We study the ground-state correlation energy E(c) of two electrons of opposite spin confined within a D-dimensional ball (D > or = 2) of radius R. In the high-density regime, we report accurate results for the exact and restricted Hartree-Fock energy, using a Hylleraas-type expansion for the former and a simple polynomial basis set for the latter. By investigating the exact limiting correlation energy E(c)(0) = lim(R-->0)E(c) for various values of D, we test our recent conjecture [P.-F. Loos and P. M. W. Gill, J. Chem. Phys. 131, 241101 (2009)] that in the large-D limit, E(c)(0) approximately -delta(2)/8 for any spherically symmetric confining external potential, where delta = 1/(D-1).

Spin-extended and configuration-interaction studies of first-row atoms

Energies and spin densities at the nucleus have been calculated by the spin-extended Hartree-Fock (SEHF) method for the ground states of Li, B, C, N, O, and F, and by configuration-interaction (CI) methods for B. The SEHF energies are only slightly lower than restricted Hartree–Fock energies. The SEHF method yields a good spin density for Li, but for the other atoms the results are strongly basis-set dependent. The CI calculations yield good correlation energies, but erratic spin densities. When applied in connection with pair-correlation methods, the CI studies indicate good additivity for one-particle and pair contributions to the energy, but not to the spin density. Valence-shell CI'S are applied to the ground states of B, B-, N, N-, O, and O-, and the results used to estimate electron affinities of B, N, and O.

Unrestricted Hartree-Fock for quantum dots

We present detailed results of unrestricted Hartree-Fock (UHF) calculations for up to eight electrons in a parabolic quantum dot. The UHF energies are shown to provide rather accurate estimates of the ground-state energy in the entire range of parameters from high densities with shell model characteristics to low densities with Wigner molecule features. To elucidate the significance of breaking the rotational symmetry, we compare restricted Hartree-Fock (RHF) and UHF calculations. While UHF symmetry breaking admits lower ground-state energies, misconceptions in the interpretation of UHF densities are pointed out. An analysis of the orbital energies shows that for very strong interaction the UHF Hamiltonian is equivalent to a tight-binding Hamiltonian. This explains why the UHF energies become nearly spin independent in this regime while the RHF energies do not. The UHF densities display an even-odd effect which is related to the angular momentum of the Wigner molecule. In a weak transversal magnetic field this even-odd effect disappears.

Asymptotic behavior of the RHF energy of the PPP model of alternant cyclic polyenes

The large N expansion of the restricted Hartree–Fock (RHF) exchange energy per atom E(N) of the Pariser–Parr–Pople (PPP) model of cyclic polyenes (annulenes) CNHN is derived in detail. We explicitly derive the coefficients E0 and E1 of the asymptotic expansion: E(N)=E0+E1 ln N/N2+O(N−2), N→∞, in the very simple case of half-filling and no bond alternation. The exchange energy per atom in the infinite chain can be written as Eex=(2/π2)∑∞j=1{[γ∞(2j−1)]/[(2j−1)2]}, where γ∞ is the two-electron repulsion integral in the infinite chain. On the other hand, the second coefficient E1 giving a finite-size correction is found to be 1/2b, where b is the bond length. This value of E1 differs slightly from that of a linear chain with periodic boundary conditions because the distance between sites depends upon the radius of the ring, i.e., upon N. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66: 397–407, 1998

Theoretical Studies of Inorganic and Organometallic Reaction Mechanisms. 10. Reversal in Stability of Rhodium and Iridium η2-Ethene and Hydridovinyl Complexes

Ab initio quantum mechanical calculations were used to examine models for the reaction [η3-HB(X)3](CO)M(η2-CH2CH2) → [η2-HB(X)3](CO)M(η2-CH2CH2) → [η3-HB(X)3](CO)M(H)(CHCH2), for which it is known that the equilibrium lies toward the hydridovinyl product for iridium, with X = 3-trifluoromethyl-5-methylpyrazol-1-yl, and lies toward the η2-ethene reactant for rhodium, with X = 3,5-dimethylpyrazol-1-yl, and most other related systems. The ligand models tested correspond to X = NHNH2, NHNCHF, N2C3H3 (pyrazol-1-yl), N2C3H2F (3-fluoropyrazol-1-yl), N2C3H2F (5-fluoropyrazol-1-yl), N2C3H2CH3 (3-methylpyrazol-1-yl), and N2C3H2CF3 (3-(trifluoromethyl)pyrazol-1-yl). For the iridium complexes, the restricted Hartree−Fock (RHF) optimized geometries are similar to the Moller−Plesset second-order (MP2) ones and the energy calculations at the MP2//RHF, MP2//MP2, and MP4SDQ//MP2 levels give reasonable results. For the rhodium complexes, although the RHF energies appear to be in qualitative accord with the calculated resul...

Correlated wavefunctions for crystalline solids: multiparameter AMO treatment of the cubic electron gas

The first successful application of the full alternant molecular orbital (AMO) method to an extended three-dimensional system is reported. The system is that of the electron gas constrained to a cubic Hartree-Fock Fermi surface. At all values of rs the AMO energy lies below the restricted Hartree-Fock energy, so an RHF to AMO transition found by Laskowski and co-workers (1973) when the AMO state was restricted to one coupling parameter is removed. This quantitative result is consistent with an inequality due to Rosengren and Johansson (1973). An unexpected additional result is that the AMO energy lies below the spherical RHF energy for rs>49 indicating the onset of a spin-polarised ground state of the uniform electron gas in rough accord with Ceperley's recent Monte Carlo calculations (1978).

The ground state of MoCr(O2CH)4 at theab initio SCF and CI levels. A symmetry adapted RHF energy functional with an artificial double minimum

Ab initio restricted Hartree-Fock (RHF) calculations carried out on the ground state of MoCr(O2CH)4 lead to two distinct energy minima according to the initial guess made for the set of trial vectors. It is shown that these two symmetry-adapted wavefunctions can be correlated with a twofold degenerate broken-symmetry solution previously characterized for the related system of higher symmetry Cr2(O2CH)4. Complete CI expansions have been carried out from either RHF polarized wavefunction using as a basis the set of eight frontier MO's with high metal character. These expansions yield poorly resymmetrized wavefunctions. A similar CI expansion has finally been carried out from a wavefunction resymmetrized at the SCF level and corresponding to a saddle point of the RHF energy hypersurface. The total energy associated with this latter expansion is the lowest obtained in the present work. The natural orbital analysis corresponds to (σ)1.86(π)3.58(δ)1.54(δ)0.46(π)0.42 (δ*)0.14 and shows that this resymmetrized CI expansion is in many respects similar to the correlated wavefunctions obtained for the homobinuclear parent systems.

Theoretical determination of molecular structure and conformation. IV. Electronic effects influencing the stability of methyl substituted primary ozonidesa),b)

The conformational surfaces of three methyl substituted primary ozonides (PO), namely, 4-methyl-1,2,3-trioxolane (propene PO), cis-4,5-dimethyl-1,2,3-trioxolane (cis-2-butene PO) and the corresponding trans PO (trans-2-butene PO), are studied with restricted Hartree–Fock (RHF) theory employing a split valence [3s2p/2s] and an augmented split valence [3s2p1d/2s] basis set. An analysis of the computed RHF energies reveals that only with polarization functions in the basis set is a reliable theoretical description of the three PO’s guaranteed. In the case of propene PO and trans-2-butene PO, reasonable results are obtained by means of a rigid pseudorotor model based on the theoretically determined structures of various 1,2,3-trioxolane conformations. However, a flexible pseudorotor model is necessary in the case of cis-2-butene PO in order to achieve a realistic account of substituent–substituent interactions. The conformational surfaces of all three PO’s closely resemble that of the parent PO, 1,2,3,-trioxolane. Thus, the preferential interconversional process of a methyl substituted PO is pseudorotation hindered by barriers between 2.3 and 3.5 kcal/mole. In marked contrast to previous semiempirical results, the oxygen envelope (E) conformations are calculated to be the most stable puckered PO forms. This finding is most important with regard to the stereochemical aspects of the ozonolysis mechanism. The methyl groups either stabilize or destabilize the puckered PO ring depending on the position of the substituent. With the aid of the pseudorotation phase angle and a well-defined substituent orientation angle, reliable predictions with regard to the electronic effect of a methyl group can be made. It is shown that only an equatorially placed methyl group leads to a favorable orbital mixing between the π-type MO’s of the ring and the substituent and a resultant lowering of the two highest occupied MO’s of the PO. The puckered ring is stabilized by back-donation of electron charge to the substituent which is not possible if this is axially positioned.

On methods for converging open-shell Hartree-Fock wave-functions

The ‘level-shifting’ technique of Saunders and Hillier is extended so as to allow guaranteed convergence of the restricted Hartree-Fock energy for many commonly found open-shell configurations. The proposed energy minimization scheme is compared with other procedures due to McWeeny, Roothaan and Segal. A test case calculation on the 2Σ+ ground state of boron oxide is described.

Electron gas test for the alternant molecular orbital method

The total energy of the uniform electron gas is calculated in the alternant molecular orbital (AMO) approximation for a cubic Fermi surface. This energy is lower than the restricted Hartree-Fock energy for the cubic Fermi surface for rs>or=9.27 au. This transition point is discussed from the point of view of Adams' instability criterion.

Z-1/2expansion of the unrestricted Hartree-Fock equations for the (1s)(1s')1S states of helium and helium-like ions: II

It is shown that the unrestricted Hartree-Fock equations may be solved by expanding the orbital wave functions in perturbation series with parameter Z-1/2 The work of a previous paper is extended to the calculation of the orbital functions to second order, and the energy to fifth order A similar calculation is made for the restricted Hartree-Fock approximation The higher-order energy coefficients are estimated Using these estimates, the restricted Hartree-Fock energy sums to -2 861 68 au, which is the value obtained by Roothaan et al, while the unrestricted Hartree-Fock energy sums to -2 877 99 au, which is the value obtained by Froese.

A Perturbation Treatment of the Unrestricted Hartree-Fock Equations for the Ground State of Lithium-like Ions

The unrestricted Hartree-Fock integro-differential equations for the ground state 1s22s 2S of lithium-like ions are solved by a perturbation expansion in ascending powers of 1/Z. The zero- and first-order functions are obtained analytically and energies up to the third order are calculated. These are compared with the corresponding expansion for the restricted Hartree-Fock energy, obtained by Linderberg (1961). Some of the difficulties in using these functions for calculating the hyperfine splitting are briefly discussed.