Hartree-Fock limit wavefunctions

It was decided to improve these calculations by using better electronic wavefunctions 0. Single configuration molecular orbital wavefunctions were still used. However, the molecular orbitals were expressed in terms of a so-called extended basis set of gaussian atomic orbitals (for details see reference (3)). The Hartree-Fock-self-consistent-field (HFSCF) procedure was carried out with the digital computer program POLYATOM, The quality of the wavefunctions is not quite what would be called Hartree-Fock limit wavefunctions. Calculations were carried out at several intemuclear distances and C was calculated with the inclusion of the factor A correctly calculated. The calculations were extended to include the ground states of several ions and also of HCl. 

Cade and Huo (13) have calculated near Hartree-Fock limit wavefunctions for LiH, BH, HF and HCl. The molecular orbital coefficients c j are available in Hie literature again at only the equilibrium Intemuclear distance. Thus again Cl values cannot be completely calculated. In Table I, the C values at the experimental equilibrium intemuclear distances calculated for LiH, BH, HF, and HCl with (1) minimum basis set wavefunctions, with (2) our extended basis set wavefunctions and with (3) the near Hartree-Fock limit wavefunctions are compared. In order to assess the quality of the various wavefunctions, the respective electronic energies are compared with those of the corresponding near Hartree-Fock limit wavefunctions. For the minimum basis set and the near Hartree-Fock limit calculations, the correct CL values of the extended basis set calculations were employed to calculate C. It is seen that both the C values and the A C values for the tended basis set calculations approach closely those of the near Hartree-Fock limit calculations. For H2, we carried out calculations not only for an extended basis set but also for a large extended basis set which 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

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Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

Unfortunately, this only holds for the exact wavefunction and certain other types ol leavefuiiction (such as at the Hartree-Fock limit). Moreover, even though the Hellmarm-Feynman forces are much easier to calculate they are very unreliable, even for accurate wavefunctions, giving rise to spurious forces (often referred to as Pulay forces [Pulay l )S7]). 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

I have dealt at length with the Hartree and the Hartree-Fock models. The father of this field, Sir William Hartree, was concerned with the atomic problem where it is routinely possible to integrate numerically the HF integro-differential equations in order to produce (numerical) wavefunctions that correspond to the Hartree-Fock limit. For molecular applications the LCAO variant of HF theory assumes a dominant role because of the reduced symmetry of the problem. 

If W were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to W is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the Bom-Oppenheimer approximation are likely to be small8 (typically 0.002 a.u.), and for most molecules relativistic effects can be ignored,6 there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. 

The second problem is the much more realistic one of the effect of a limited basis set expansion. This is clearly a more serious problem because only for linear molecules or those with a few first-row atoms can the Hartree-Fock limit be reached at present. For many of the molecules with which theoretical chemists deal, wave-functions of such accuracy are not available but it may be some comfort to know that even if they were they need not give very good answers It should be mentioned that Brillouin s theorem applies to any SCF wavefunction, but unless the wavefunction is near the Hartree-Fock limit the electron distribution cannot be expected to be a close representation of the true one. No general treatment of this problem has been given neither does one seem possible since it would depend on the ways in which the basis set under consideration was weak, and these may be many. 

For any variational wavefunction which is not near the Hartree-Fock limit the Brillouin theorem is irrelevant, and even for those of Hartree-Fock accuracy low-lying important excited states may invalidate the conclusions drawn from it. The statement that values of one-electron properties are expected to be good because of the Brillouin theorem should therefore be regarded with caution. 

Using a large basis set of GTO s, Meyer and Pulay68 obtained theoretical estimates of the symmetry force constants for CH4 which differed from the experimental values by amounts ranging from 0 to 12%. They concluded that the values obtained at the true Hartree-Fock limit would deviate by no more than 5% from the experimental ones. A similar approach, for use with extended Huckel and MINDO wavefunctions has been developed by Mclver and Komomicki.69... 

Hartree-Fock Potential Energy Surface Calculations Asymptotic Behaviour of Single Determinantal Wavefunctions.—In general, a single determinantal wavefunction, whether or not it is at the Hartree-Fock limit, does not provide an adequate description of a molecular system over the complete range of intemuclear separations, because of the failure of such a function to describe... 

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

Theoretical studies of PO have been carried out recently by Mulliken and Liu,283 who obtained a wavefunction close to the Hartree-Fock limit. An investigation of the Rydberg states has also been reported.335 The first calculations using Cl were those of Tseng and Grein,386 who have studied a variety of PE curves for the low-lying 2II, 4II, 2E+, 2S, 4S-, 2A, and 2fl> states. A minimal STO basis and full valence-shell Cl were used. The relative positions of the states agree well with experiment, and several predictions were made for as yet unobserved states. 

Finally there will be an error in At if the wavefunction is not at the Hartree-Fock limit. If the correction terms are roughly constant for different i, Koopmans theorem may still predict the correct ordering of the photoelectron peaks. If it fails,... 

When the basis set contains many terms (effectively infinite), one obtains the best possible result from solution of the Hartree-Fock equation the Hartree-Fock limit. Improvements beyond this limit are most usually achieved by allowing the molecular wavefunction I7 to be a linear combination of antisymmetrized products of orbitals i,... 

As the wavefunction approaches the Hartree-Fock limit one would expect the Ti-Cl bond distance to be shorter than the experiment because of the lack of bond-pair correlation. The bond-pair correlation added by the GVB wavefunction lengthened the Ti-Cl bond 0.021 A, because the GVB wavefunction adds only limited left-right correlation and none of the dynamical correlation. For most A-B bonds, the calculated bond lengths at the SCF level are too short, and the correlation added by a GVB calculation accounts for a major portion of the non-dynamical correlation error in the SCF wavefunction. But for Ti-Cl bonds, both the SCF and GVB calculations predict too long a bond distance because they do not include necessary dynamical atomic correlation of the Cl atoms. 

Optimization of the geometry of TiCU at the SCF level results in a Ti-Cl bond length which is longer than the experiment, even when d- and f-type polarization functions are added to the basis set. For covalently bonded systems one expects a wavefunction at the Hartree-Fock limit to give bond lengths shorter than the experiment if they are not sterically crowded. Because the Hartree-Fock wavefunction overestimates the Cl -Cl repulsions, the Ti-Cl bond distances remain long, even in large basis sets. 

It is found that, in general, the molecular charge densities obtained from single determinant SCF calculations carried essentially to the Hartree-Fock limit (see page 163) are accurate to within 2-5% of the total charge density at any point in space, except perhaps for very large distances from the nuclei. Coulomb electron correlation effects are relatively small in particular, the value of p(r) in the neighbourhood of a bond critical point calculated in that way exceeds the value obtained from a correlated wavefunction by only a few per cent. 

We now use a theorem, proved elsewhere , that A = 0 S% hence small, not only at the Hartree-Fock limit but also when and i/r are SCF wavefunctions obtained in a finite common (dimer) basis set. Suppose we employ for this purpose the basis (Xa Xb ) This leaves and hence... 

The most precisely defined quantity is the correlation energy, the difference between the energy of a molecule calculated at the Hartree-Fock limit and the energy that would result from an exact solution of the nonrelativistic Schrodinger equation. This energy difference corresponds to dynamic correlation, which may be included in an accurate wavefunction by use of terms in... 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

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