The Hartree-Fock Limit

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. 

A more useful quantity for comparison with experiment is the heat of formation, which is defined as the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states. The heat of formation can thus be calculated by subtracting the heats of atomisation of the elements and the atomic ionisation energies from the total energy. Unfortunately, ab initio calculations that do not include electron correlation (which we will discuss in Chapter 3) provide uniformly poor estimates of heats of formation w ith errors in bond dissociation energies of 25-40 kcal/mol, even at the Hartree-Fock limit for diatomic molecules. 

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]). 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

The first cell in the last tow of the table represents the Hartree-Fock limit the best approximation that can be achieved without taking electron correlation into account. Its location on the chart is rather far from the exact solution. Although in some cases, quite good results can be achieved with Hartree-Fock theory alone, in many others, its performance ranges from orfly fair to quite poor. We ll look at some these cases in Chapters 5 and 6. 

Solution of the numerical HF equations to full accuracy is routine in the case of atoms. We say that such calculations are at the Hartree-Fock limit. These represent the best solution possible within the orbital model. For large molecules, solutions at the HF limit are not possible, which brings me to my next topic. 

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

Owing to their simplicity, the helium atom and the dihydrogen molecnle have been the object of experiments (Ref. 31 for a and 7 of He Ref. 32 for a of H2) and calcnlations, some of them near the Hartree-Fock limit (Ref. 33 for He and Ref. 34-36 forH2 ). In order to test our polarization fnnctions, we have taken the zeroth... 

The theoretical results provided by the large basis sets II-V are much smaller than those from previous references [15-18] the present findings confirm that the second-hyperpolarizability is largely affected by the basis set characteristics. It is very difficult to assess the accuracy of a given CHF calculation of 2(ap iS, and it may well happen that smaller basis sets provide theoretical values of apparently better quality. Whereas the diagonal eomponents of the eleetrie dipole polarizability are quadratic properties for which the Hartree-Fock limit can be estimated with relative accuracy a posteriori, e.g., via extended calculations [38], it does not seem possible to establish a variational principle for, and/or upper and lower bounds to, either and atris-... 

Until recently, only estimates of the Hartree-Fock limit were available for molecular systems. Now, finite difference [16-24] and finite element [25-28] calculations can yield Hartree-Fock energies for diatomic molecules to at least the 1 ghartree level of accuracy and, furthermore, the ubiquitous finite basis set approach can be developed so as to approach this level of accuracy [29,30] whilst also supporting a representation of the whole one-electron spectrum which is an essential ingredient of subsequent correlation treatments. 

As we said in the introduction, the only consistent framework for a relativistic many-electron system is QED. By means of the Hartree-Fock limit of this theory, after renormalization, and using gradient techniques, Engel and Dreizler [22] found a complete energy functional where both terms of the two previous sections appear naturally. 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

The Hartree-Fock limit for the ground state of the hydrogen molecule can be achieved in principle by using a basis set of s-type Gaussian functions distributed along the internuclear axis. Such a basis set will be termed an axis basis set. The atom-centred basis functions, designated (ac), are located on the points... 

The vertical axis designates the basis set. At the top is a so-called minimal basis set, which involves the fewest possible functions (see discussion later in this chapter), while at the very bottom is a complete basis set. The bottom of the column of Hartree-Fock models (at the far left) is termed the Hartree-Fock limit. Note, that this limit is not the same as the exact solution of the Schrodinger equation (or experiment). 

Limiting (6-311+G basis set) Hartree-Fock bond lengths in the compounds examined are consistently shorter than experimental distances, the same result as seen for multiple bonds in hydrocarbons. Results obtained using the 6-31G basis set are nearly identical, which suggests that the 6-31G model closely reflects the Hartree-Fock limit insofar as bond length calculations for these types of systems. 

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