Corrected Hartree functionals

We have thus arrived at an approximate NOE that coincides with the selfinteraction-corrected Hartree functional proposed by Goedecker and Umrigar except for the choice of phases given by the sign of (Ap. Unfortunately, this NOF gives a wrong description of the ONs for the lowest occupied orbitals. In order to ensure that these ONs only are close to unity, we propose to add a new term to the functional form (94) of matrix A, namely,... 

As it is now very well known, accurate studies of the water-water interaction by means of ab-initio techniques require the use of larger and flexible basis sets and methods which consider correlation effects [85,94-96], Since high level ab-initio post-Hartree-Fock calculations are unfeasible because of their high computational cost for systems with many degrees of freedom, Density Functional Theory, more economical from the computational point of view, is being more and more considered as a viable alternative. Recently, we have presented [97] results of structural parameters and vibrational frequencies for the water clusters (H20) , n=2 to 8, using the DFT method with gradient corrected density functionals. 

Although in principle an exact solution to the Schrodinger equation can be expressed in the form of equation (A.13), the wave functions and coefficients da cannot to determined for an infinitely large set. In the Hartree-Fock approximation, it is assumed that the summation in equation (A.13) may be approximated by a single term, that is, that the correct wave function may be approximated by a single determinantal wave function , the first term of equation (A.13). The method of variations is used to determine the... 

We have in this chapter discussed different situations, where the simple single-configurational Hartree-Fock description of the molecular system breaks down. It has been shown that a model can be devised that gives a qualitatively correct wave function in cases where several electronic configurations are close in energy. In the next chapter we shall develop the tools needed to compute such wave functions. 

Cohen AJ, Baerends EJ (2002) Variational density matrix functional calculations for the corrected Hartree and corrected Hartree-Fock functionals, Chem. Phys. Lett, 364 409—419... 

Our earlier discussion of electronic wave functions for many-electron atoms drew attention to the main inadequacy of the Hartree-Fock single determinant treatment it does not take account of the correlation between the motions of electrons with opposite spins. In molecules this can even lead to qualitative deficiencies in the description of electronic structure, such as the failure to describe dissociation correctly. For example, the correct wave function for the singlet state of the hydrogen molecule at large... 

Iq this study the B3LYP hybrid method proposed by Becke [31] and included in the Gaussian 98 [30] package was used. This method includes a mixture of Hartree-Fock and DFT exchange terms associated with the gradient corrected correlation functional of Lee et al [32], In all the calculations the metal cluster geometry was kept frozen. 

If a -electron wave function is limited to a Slater determinant of n spin orbitals, one stays within the frame of the independent-particle model, and the best model of that sort (for a discussion, see 22>) for a given problem is that in which the orbitals used to construct the wave function are solutions of the Hartree-Fock equations. This model is only an approximation of the correct wave function. As mentioned in Sect. 3.1, the wave function should be written as a linear combination of Slater determinants, as in Eq. (3.4). To illustrate this, let us consider a two-electron system where the spin can be separated off, so that it is sufficient to consider a function ip (1,2) depending only on the space coordinates of the two particles 1 and 2. For a singlet state ip (1,2) is symmetric with respect to space coordinates ... 

Since no bonds are broken in the dissociation of a van der Waals molecule, the Hartree-Fock determinant is a correct reference function for, e.g., a CEPA calculation. As Werner and Meyer show, a CEPA calculation is able to reproduce experimental dipole moments and polarizabilities very well, provided a sufficiently large basis is employed. The electrostatic and induction components are thus well taken care of The only remaining problem is the dispersion energy. The CEPA method, as a pair method, should be quite good... 

Dynamical electron correlation effects are intrinsically included in DFT methods via the local and gradient-corrected correlation functionals. Most importantly, the nondynamlcal electron correlation effects have been partially accounted for via the exchange functionals (33,34). As a consequence, the multiconfiguration characters that often show up in traditional ab initio calculations using Hartree-Fock references could be sUghtly reduced in DFT methods. In other words, for approximating the true electronic stmctures of many-electron atoms and molecules... 

Solution of KS equations at the level of local or gradient-corrected density functionals always requires calculation of mathematically complicated integrals. Unlike the exchange integrals of Hartree Fock quantum mechanics, which can be solved analytically for Gaussians, the density functional exchange and correlation approach introduces integrals which are usually more involved than the matrix element < 4>iy-lZci >= C / 1(r)/>1/3(r)< (r)d3r. 

One major source of error in Eq. (7) or AGf is the calculation of the O2 molecule using single determinant theories of electronic structure (such as DFT and Hartree-Fock) [80,81]. In general, gradient-corrected DFT functionals overestimate formation and binding energies and it is estimated that these are accurate for O2 to less than 0.5 eV [48]. It is fortunate however that many studies of stirface stability are not sensitive even to this magnitude of discrepancy in po, as is clear in the examples discussed below. 

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