Approximations , Adiabatic Hartree-Fock

Further we can proceed similarly as in the case of adiabatic approximation. We shall not present here the details, these are presented in [21,22]. We just mention the most important features of our transformation (46-50). Firstly, when passing from crude adiabatic to adiabatic approximation the force constant changed from second derivative of electron-nuclei interaction ufcF to second derivative of Hartree-Fock energy Therefore when performing transformation (46-50) we expect change offeree constant and therefore change of the vibrational part of Hamiltonian... 

One reason for interest in more accurate calculations on HF has been the measured dissociation energy Z)o(HF), which can be obtained from photoionization or photoelectron spectra. Since HF+ dissociates correctly within the Hartree-Fock approximation to H+ and F(2P) in both the X2Tl and 2S+ states, PE curves were calculated by Julienne et a/.,203 and later by Bondybey et a/.,187 both of whom obtained values of Do in good agreement with experiment for the SCF calculation. The bond length in HF+ of 1.000 bohr is less than the experimental result, which the authors call into question. Julienne et al. give a detailed discussion of the adiabatic dissociation process. HF+ was also considered in the calculations of ref. 198. Richards and co-workers have reported several calculations of the PE curves of HF+, including the A2E+ state, which is correctly predicted to be bound.204... 

The quantity ffffr.r. a ) is the frequency-dependent XC kernel for which common approximations are applied frequency-independent (adiabatic) local density approximations (LDA), adiabatic generalized gradient approximations (GGA), hybrid-DFT variants such as the popular functionals B3LYP and PBEO in which Kxc contains an admixture of Hartree-Fock ( exact ) exchange X,... 

The adiabatic potential energy curves for these electronic states calculated in the Born-Oppenhelmer approximation, are given in Figure 1. Since we have discussed the choice of basis functions and the choice of configurations for these multiconfiguration self-consistent field (MCSCF) computations (12) previously (] - ), we shall not explore these questions in any detail here. Suffice it to say that the basis set for Li describes the lowest 2s and 2p states of the Li atom at essentially the Hartree-Fock level of accuracy, and includes a set of crudely optimized d functions to accommodate molecular polarization effects. The basis set we employed for calculations involving Na is somewhat less well optimized than is the Li basis in particular, so molecular orbitals are not as well described for Na2 (relatively speaking) as they are for LI2. 

The idea of mixing density functional approximations with exact (Hartree-Fock-Uke) exchange rests on theoretical considerations involving the adiabatic connection formula... 

At present DFT calculations for frequency dependent properties are not competitive with conventional Hartree-Fock based ones as the methods are not fully developed. There is also much work still to be done to develop suitable functionals, probably current dependent ones, and the theory beyond the adiabatic approximation needs much more investigation. Nevertheless the initial results are promising, and the relative... 

The final form of the Born-Handy formula consists of three terms The first one represents the electron-vibrational interaction. I will not present the numerical details for H2, HD and D2 molecules here, it can be found in our previous work. The most important result here is that the electron-vibrational Hamiltonian is totally inadequate for the description of the adiabatic correction to the molecular groundstates its contribution differs almost in one decimal place from the real values acquired from the Born-Handy formula. In the case of concrete examples -H2, HD and D2 molecules - the first term contributes only with ca 20% of the total value. The dominant rest - 80% of the total contribution - depends of the electron-translational and electron-rotational interaction [22]. This interesting effect occurs on the one-particle level, and it justifies the use of one-determinant expansion of the wave function (28.2). Of course, we can calculate the corrections beyond the Hartree-Fock approximation by means of many-body perturbation theory, as it was done in our work [22], but at this moment it is irrelevant to further considerations. 

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