Hartree-Fock function transformation

FIGURE 3.4 Transformation of the valence orbitals of BeH2, from canonical MOs (left-hand side) to localized bond orbitals (right-hand side). This transformation leaves the polyelectronic Hartree-Fock function unchanged. 

If the Hartree-Fock functions f undergo a transformation H ifu = u, and all the functions up are situated in the domain of u, then... 

The Fock matrix is now recalculated from the transformed MOs and an iterative process is thus established, in which the Fock matrix is repeatedly constructed and diagonalized until the MOs from which it is constructed are the same as those generated by its diagonalization. At this stage, we have satisfied the canonical conditions (10.3.16) and the solution is said to be self-consistent. In the same parlance, the field generated by the converged Fock potential is said to be self-consistent and the Hartree-Fock function itself is called a self-consistent field (SCF) 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. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

In Fig. 5, we present plots for the transformation function /(r), for the Jacobian of the transformation J(f (r) r), and for the derivative of the transformation function, df r)fdr corresponding to the example at hand. Selected values for the transformed orbitals are presented in Table 1, where we have also listed the corresponding values of the Hartree-Fock-Raffenetti orbitals, as well as the difference between untransformed and transformed orbitals for the sake of completeness. 

Table I. Selected values of the Raffenetti-Hartree-Fock orbitals Isg and 2shf for Be, of their locally-scaled transformed functions IsJ, and 2sgr and of their differenees di, = Ish, - Isgj,) and = (2siif — 2sgp). [Reproduced with permission from Table I Ludeiia et al. [Ill]]...

In Table VIII, we present the local-scaling- transformation-energy results for lithium and beryllium and compare them with results obtained with other methods. It is worth mentioning that the Hartree-Fock results for these atoms are a first instance of atxurate energy values obtained within the context of a formalism based on density functional theory. 

We review in this Section some recent work by Ludena, Lopez-Boada and Pino [113] on the construction of energy functionals that depend explicitly upon the one-particle density, but which are generated in the context of the local-scaling-transformation version of density functional theory. This work does not consider the general case involving exchange and correlation, but restricts itself to the exchange-only Hartree-Fock approximation. 

For the Hartree-Fock case, the energy functional [4> >] for the density-transformed single Slater determinant given by Eq. (152), may be rewritten as a functional of the one-particle density ... 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

The reader should note that no transformation operator 0A can alter the radial function J r) of an orbital and consequently the symmetry properties of the AOs are completely defined by the angular functions, Y O, ). Since these angular functions are the same in all one-electron product function approximations, the orbitals in all these approximations (Slater orbitals, numerical Hartree-Fock orbitals,... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

Analysis of the properties of the transformed radial orbitals testify that these functions are a realistic practical alternative to the solutions of multiconfiguration (Hartree-Fock-Jucys) equations. Moreover, the generation of solutions using the code is fairly simple and not computer time consuming even for heavy atoms and ions. 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

In the second place, a quite useful characteristic of LS-DFT is that it renders possible to transform an arbitrary wavefunction, say, the Hartree-Fock single Slater determinant into a locally-scaled one associated with a given one-particle density such as the exact one. Thus, one can easily generate a locally-scaled Hartree-Fock wavefunction that yields the exact p. In this sense, one finds much common ground between LS-DFT and those constructive realizations of the constrained-search approach which reformulate the Hartree-Fock method as well as with those developments which pose the optimized potential method as a particular instance of density functional theory [42,43,57-61]. 

R. Lopez-Boada, R. Pino and E.V. Ludena. Hartree-Fock energy density functionals generated by local-scaling transformations Applications to first-row atoms. J. Chem. Phys. (submitted). 

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