Hartree repulsive potential

The Hartree LR function of only the Hartree repulsion potential is given by... 

If this is combined with the expression for the nncIeMlectron attractive potential and the electron-electron Hartree repulsive potential we have the TF expression for the energy of a homogeneous gas of electrons in a given external potential ... 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

Both Hartree-Fock and Wolfsberg-Helmholz and related methods have been applied to metal carbonyl compounds. The Hartree-Fock method 90) treats each electron as moving in a potential field due to the nuclei and to the charge-density of all the other electrons. The interelectronic repulsion potential takes account of the tendency of electrons of parallel spin to avoid each other (exclusion), but not of correlation due to electron-electron repulsion independent of spin ). The field due to some assumed electron distribution is evaluated, and wave-functions calculated for motion in that field. When sufficiently close agreement between assumed and calculated wave-functions is finally obtained, the solution is said to be self-consistent . 

In the 1930s Douglas Hartree and his father William Hartree used a mechanical analog computer to explore the idea that an electron in an atom moves partly in the attractive potential of the nucleus and partly in an averaged repulsive potential due to all the other electrons. Later, V. Fock added to the Hartrees model an exchange term due to the effects of the antisymmetry of the many-electron wave function. The Hartree-Fock approximation is still a basic tool of quantum chemistry. 

For a derivation of the Hartree-Fock equations, see, e.g., [10].) The first term is the kinetic energy of electron i. The second term is the potential energy in Vext, which is the Coulomb potential due to the nucleus. The third term is a correction to the Coulomb potential due to the centrally symmetric average of the electrostatic repulsion between the electrons. The fourth term is an effective exchange potential, due to the Pauli Principle. The sum of the external, Coulomb and electrostatic repulsion potentials is often referred to as the Hartree potential. 

These are the Hartree-Fock equations. The first summation term (the coulomb potential) is the repulsive potential experienced by an electron in orbital j at ri due to the presence of all the other electrons in orbitals k at r2. Note however that this summation also contains a term corresponding to an electron s interaction with itself (i.e., when j=k) and this self-interaction must be compensated for. The second summation is called the exchange potential. The exchange potential modifies the interelectronic repulsion between electrons with like spin. Because no two electrons with the same spin can be in the same orbital j, the exchange term removes those interactions from the coulombic potential field. The exchange term arises entirely because of the antisymmetry of the determinental wavefunctions. The exchange term also acts to perform the self-interaction correction since it is equal in magnitude to the coulomb term when j =k. 

Slater s derivation Previously we found that in the Hartree-Fock equation, the electron moves in the repulsive field of all electrons, including itself, but with density subtracted corresponding to one missing electron around it. We assume that the density is constant Slater derived the radius of the hole (rj and then calculated the decrease in the repulsive potential because of the hole. The following relation must hold ... 

The first three terms are similar to HF theory, thus corresponding to the kinetic energy of the electron, the potential for nuclear-electron attractive interactions and the Hartree repulsive interactions between electrons. The final term, Yxcir), corresponds to the exchange correlation potential which is the derivative of the exchange correlation energy with respect to the density. This is more formally recognized as the chemical potential and written as... 

Finally, we should also briefly discuss the performance of semiempirical methods. These are methods that neglect some of the more expensive integrals in Hartree-Fock molecular orbital theory and replace others with empirical parameters. Because semiempirical methods are based on Hartree-Fock theory, and because Hartree-Fock theory does not capture dispersion effects, semiempirical methods are not suitable for computing dispersion-dominated noncovalent interactions. Semiempirical methods yield repulsive potentials for the sandwich benzene dimer, just as Hartree-Fock does. However, given that semiempirical methods already contain empirical parameters, there is no reason not to fix this deficiency by adding terms proportional to r, as is done in force-field methods and the empirical DFT-D methods. Such an approach has been tested for some base pairs and sulfur-7t model systems. 

There we see once again that the SCF approximation yields a barrier height much larger than experiment. And in fact even if one goes to a complete set of one-electron functions, the Hartree-Fock limit barrier height will be . 25 kcal larger than the true barrier. This inherent inability of the SCF approximation to even qualitatively describe repulsive potential surfaces must be viewed as one of the most important developments of our research to date. 

The M0ller-Plesset perturbation method was introduced around 1975. It differs from the ordinary perturbation method introduced in Chapter 19 in that the unperturbed wave function is taken to be the Hartree-Fock wave function without configuration interaction. The perturbation term is taken as the difference between the Hartree-Fock potential and the actual interelectron repulsion potential.Calculations are usually carried out to second order but calculations to the fourth order have been done. 

Since the dispersion interaction cannot be recovered at the unctMTelated Hartree-Fock level, the Hartree-Fock potential-energy curve should be repulsive for all intemuclear separations. Nevertheless, we note from Figure 8.23 that, for several of the basis sets considered here, the uncorrected... 

Hartree-Fock interaction is attractive at the experimental minimum. The origin of this interaction is BSSE, whose spurious stabilization of the neon dimer outweighs the Pauli repulsion for small basis sets. These effects are illustrated on the left in Figure 8.24, where we have plotted the (uncorrected) Hartree-Fock potential-energy curve for various doubly augmented basis sets. As the cardinal number increases, the spurious Hartree-Fock minimum becomes shallower and located further out. We note that the potential-energy curve may contain several such spurious minima. Thus, the d-aug-cc-pVDZ curve has a minimum of —73 pEh at a separation of 6.1oo a shallower minimum of —14 pEh at a separation of 8.5ao-... 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

---