The Hartree Theory

In determinant form, the eigenfunction for an atom with three electrons is 

The fact that the sign of a determinant changes when any two columns are interchanged ensures that the condition of the Pauli principle holds. By analogy, for N electrons the antisymmetrical expression can be written compactly as the N x N Slater determinant 

The Schrodinger equation for a multi-electron atom can be written in the form 

The first term on the left-hand side of (3.27) is the sum of kinetic energies of individual electrons. —l/(4jteo) J2r Ze / r — lt is the potential energy of the interaction of tth electron with aU nuclei R. The second term —(l)/(4jifo) 

Ze /I fj — U represents potential energy of interaction of all electrons with all nuclei. The last term describes the Coulomb electron-electron interactions. Calculation of this term presents great difficulties. We must consider the Coulomb interactions between each electron and aU other electrons in the atom. The determination of the ground-state eigenfunction and the ground-state energy of a quantum many-body system appears to be a formidable problem. 

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Chapter 3 is devoted to atoms. One-electron atom and multi-electron atoms of chemical elements are considered. The probability density functions (orbitals) for electrons are illustrated. The Hartree theory is presented as a first method of approximation that has been proposed in order to calculate wave functions and energies of electrons in atoms. The covalently bonded diatomic molecules are subject of the consequent consideration. 

D.R. Hartree and V.A. Fock made an important contribution to the science of systems consisting of a large number of electrons and nuclei. We review the Hartree theory, which gives an approximation method for the determination of the ground-state eigenfunction and the corresponding eigenvalue of a many-body quantum system. 

The Hartree theory involves solving (3.28) for a system of Z electrons moving independently in the atom. The total potential of the atom can be written as a sum of a set of Z identical potentials V( rj), each depending on the radial coordinate r of one electron only. Consequently, the equation can be separated into a set of Z time-... 

Here we consider the results of the Hartree theory of multi-electron atoms given in [2]. 

Figure 3.10 presents the radial behavior of the multi-electron atom eigenfunctions for an argon atom, 2 = 18. The innermost electron shell Is has a much smaller radius than the hydrogen shell because the ratio of their nuclei charges is 18 1. For argon, the radius of the innermost shell (n = 1) is 0.07 ao- The Hartree theory predicts that the radius of the n = 1 shell for multi-electron atoms is smaller than that of the n = 1 shell for the one-electron atoms by the factor 1/(2 — 2). In multi-electron atoms, the inner shells of n = 1, n = 2 have relatively small radii because for these shells there is little shielding, and the electrons feel the full Coulomb attraction. 

The basis of the Kohn-Sham functional is an approximation to exchange-correlation energy of electrons. The approximations should include all many-body contributions to energy that is beyond the Hartree theory. The most common choices for the exchangeenergy functionals are local density approximation and generalized density approximation (Section 8.6). However, application of these functionals reveals a poor fitness for treatment systems that are bonded by the van der Waals forces. 

Thus, when expressing matrix elements of the many-body wavefunction in the position representation, the set of variables appearing as arguments in the single-particle states of the bra and the ket must be in exactly the same order for example, in the Hartree theory, Eq. (2.10), the Coulomb repulsion term is represented by... 

Thus within the Hartree theory we get the same energies as in the zero potential case. The Hartree-Fock case can now be treated by adding the exchange term to... 

A fiirther diflfieulty arises beeause the exaet wavefiinetions of the isolated moleeules are not known, exeept for one-eleetron systems. A eoimnon starting point is the Hartree-Foek wavefiinetions of the individual moleeules. It is then neeessary to inelude the eflfeets of intramoleeular eleetron eorrelation by eonsidering them as additional perturbations. Jeziorski and eoworkers [M] have developed and eomputationally implemented a triple perturbation theory of the syimnetry-adapted type. They have applied their method, dubbed SAPT, to many interaetions with more sueeess than might have been expeeted given the fiindamental doubts raised about the method. SAPT is eurrently both usefiil and praetieal. A reeent applieation [ ] to the CO2 dimer is illustrative of what ean be aehieved widi SAPT, and a rieh soiiree of referenees to previous SAPT work. 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

A textbook describing the theory associated with calculation s of Ih e electronic structure of molecti lar system s. While the book focuses on ab ini/rci calculation s, much of the in formation is also relevant to semi-empirical methods. The sections on the Hartree-fock an d Con figuration ItUeracTion s tn elh ods, in particular, apply to HyperChem. fhe self-paced exercisesare useful for the beginning computational chemist. 

To obtain an improvement on the Hartree-Fock energy it is therefore necessary to use Moller-Plesset perturbation theory to at least second order. This level of theory is referred to as MP2 and involves the integral J dr. The higher-order wavefunction g is... 

Within the periodic Hartree-Fock approach it is possible to incorporate many of the variants that we have discussed, such as LFHF or RHF. Density functional theory can also be used. I his makes it possible to compare the results obtained from these variants. Whilst density functional theory is more widely used for solid-state applications, there are certain types of problem that are currently more amenable to the Hartree-Fock method. Of particular ii. Icvance here are systems containing unpaired electrons, two recent examples being the clci tronic and magnetic properties of nickel oxide and alkaline earth oxides doped with alkali metal ions (Li in CaO) [Dovesi et al. 2000]. 

The Seetion on More Quantitive Aspects of Electronic Structure Calculations introduees many of the eomputational ehemistry methods that are used to quantitatively evaluate moleeular orbital and eonfiguration mixing amplitudes. The Hartree-Foek self-eonsistent field (SCF), eonfiguration interaetion (Cl), multieonfigurational SCF (MCSCF), many-body and Moller-Plesset perturbation theories. 

Correlation can be added as a perturbation from the Hartree-Fock wave function. This is called Moller-Plesset perturbation theory. In mapping the HF wave function onto a perturbation theory formulation, HF becomes a hrst-order perturbation. Thus, a minimal amount of correlation is added by using the second-order MP2 method. Third-order (MP3) and fourth-order (MP4) calculations are also common. The accuracy of an MP4 calculation is roughly equivalent to the accuracy of a CISD calculation. MP5 and higher calculations are seldom done due to the high computational cost (A time complexity or worse). 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

Configuration interaction (Cl) is a systematic procedure for going beyond the Hartree-Fock approximation. A different systematic approach for finding the correlation energy is perturbation theory... 

Single point energy calculations can be performed at any level of theory and with small or large basis sets. The ones we ll do in this chapter will be at the Hartree-Fock level with medium-sized basis sets, but keep in mind that high accuracy energy computations are set up and interpreted in very much the same way. 

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