Hartree type wave function

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

The electron densities of the separate atoms are described by Hartree-Fock wave functions approximated by analytic extended (Slater-type) basis sets. 

A second example is the minimal-basis-set (MBS) Hartree-Fock wave function for the diatomic molecule hydrogen fluoride, HF (Ransil 1960). The basis orbitals are six Slater-type (i.e., single exponential) functions, one for each inner and valence shell orbital of the two atoms. They are the Is function on the hydrogen atom, and the Is, 2s, 2per, and two 2pn functions on the fluorine atom. The 2sF function is an exponential function to which a term is added that introduces the radial node, and ensures orthogonality with the Is function on fluorine. To indicate the orthogonality, it is labeled 2s F. The orbital is described by... 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

Calculations based on the continuum dielectric model have been performed by the hydrated electron in the limit of zero cavity size (19). The general treatment is based on a variational calculation using hydrogenic type wave functions for the ground and the first excited states. This treatment is based on a Hartree Fock scheme, where the Coulomb and exchange interaction of the excess electron with the medium are replaced by the polarization energy of a continuous dielectric. The results obtained are summarized in Table V. The fair agreement obtained with... 

DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with j3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Rp.strirted Open-shell Hartree-Fock (RQHF). For open-shell species a UHF treatment... 

Figure 8 Second order correlation energy diagrams for a closed-shell system described in zero order by a single determinantal Hartree-Fock wave function constructed from canonical orbitals. The one Brandow diagram of this type is shown in (a). The exchange diagrams are shown in Goldstone form in (b)...

To overcome the deficiencies of the Hartree-Fock wave function (for example, improper behavior R oo and incorrect values), one can introduce configuration interaction (Cl), thus going beyond the Hartree-Fock approximation. Recall (Section 11.3) that in a molecular Cl calculation one begins with a set of basis functions Xi, does an SCF calculation to find SCf occupied and virtual (unoccupied) MOs, uses these MOs to form configuration (state) functions writes the molecular wave function i/ as a linear combination 2/ of the configuration functions, and uses the variation method to find the ft, s. In calculations on diatomic molecules, the basis functions can be Slater-type AOs, some centered on one atom, the remainder on the second atom. 

Since geometry derivatives are important for optimizing geometries, it may be useful to look in more detail at the quantities involved in calculating first and second derivatives of a Hartree-Fock wave function with a Gaussian type basis set, with the expressions for density functional methods being very similar. These formulas are most easily derived directly from the HF energy expressed in terms of the atomic quantities (eq. (3.54)). ... 

So fiir in this chapter we have discussed the Hartree-Fock equations from a formal point of view in terms of a general set of spin orbitals xj. We are now in a position to consider the actual calculation of Hartree-Fock wave functions, and we must be more specific about the form of the spin orbitals. In the last chapter we briefly discussed two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a (spin up) and jS (spin down) spin functions and unrestricted spin orbitals, which have different spatial functions for a and P spins. Later in this chapter we will discuss the unrestricted Hartree-Fock formalism and unrestricted calculations. In this section we are concerned with procedures for calculating restricted Hartree-Fock wave functions and, specifically, we consider here... 

For SCF calculations on diatomic molecules, one can use Slater-type orbitals [Eq. (11.14)] centered on the various atoms of the molecule as the basis functions. (For an alternative choice, see Section 15.4.) The procedure used to find the coefficients Cj, of the basis functions in each SCF MO is discussed in Section 14.3. To have a complete set of AO basis functions, an infinite number of Slater orbitals are needed, but the true molecular Hartree-Fock wave function can be closely approximated with a reasonably small number of carefully chosen Slater orbitals. A minimal basis set for a molecular SCF calculation consists of a single basis function for each inner-sheU AO and each valence-shell AO of each atom. An extended basis set is a set that is larger than a minimal set. Minimal-basis-set SCF calculations are easier than extended-basis-set calculations, but the latter are much more accurate. 

At this point the reader may be wondering where it all ends. In theory, the answer is never. To construct a complete basis set, capable of exactly representing the Hartree-Fock wave function for any molecule, it would be necessary to include an infinite number of functions of each symmetry type (s,p,d,f,. ..). This is sometimes referred to as the Hartree-Fock limit. For an in-depth examination of this issue the reader is referred to representative work by McDowelP and Klahn. Although a rigorous examination of completeness is beyond the scope of the present treatment, it is helpful to consider a more practical definition of completeness that allows for real world limitations. We thus arrive at the notion of effectively complete basis sets. 

In this section we shall introduce the very useful concept of the Fermi vacuum, which makes the evaluation of certain types of matrix elements much easier. As a matter of fact, many (if not most) quantum-chemical considerations and methods are based on the Hartree-Fock single determinantal wave function which serves also as a zeroth-order wave function ( reference state ) in guessing more accurate wave functions as well. For this reason, one is often interested in evaluating expectation values with respect to Hartree-Fock-type wave functions. The evaluation of such expressions will be analyzed below in some detail. 

Derivations of the above type can be made much simpler by introducing the Fermi vacuum. Consider the expectation value of some operator A with a Hartree-Fock-type wave function ... 

In this section we have presented a new derivation scheme for gradient formulae for all types of variational wave functions which violate the Hellmann-Feynman theorem only as a consequence of using an incomplete basis set. We have pointed out that if the second quantized Hamiltonian is applied, the Hellmann-Feynman theorem formally holds even in a finite basis, and the first derivatives of the energy can be obtained without considering wave function forces explicitly. On this ground, we derived a unified gradient formula from which the known results can be recovered in every special case. This has been demonstrated for the case of the Hartree-Fock wave function. 

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