Hartree-Fock function canonical

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. 

Let us first evaluate the canonical Hartree-Fock functions = Vj, (r2,. .. a tn which are solutions to the equations... 

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. 

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... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

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 many-body perturbation theory [39] [40] [41] was used to model the electronic structure of the atomic systems studied in this work. The theory developed with respect to a Hartree-Fock reference function constructed from canonical orbitals is employed. This formulation is numerically equivalent to the M ler-Plesset theory[42] [43]. 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

The structures of the model compounds C40H20 (A) and C60H12 (B) (Chart 2.1) were optimized with quantum mechanical calculations at various levels of theory (semiempirical, Hartree-Fock and hybrid density functional methods) and were found to reproduce the experimental data very well (within ca. 1.5%, Table 2.1). The optimized structures and experimental structures of A and B also match well with the structure obtained from an estimation of the Pauli bond order [34] assuming equal contributions of all 125 canonical resonance structures. 

Now we are ready to start the derivation of the intermediate scheme bridging quantum and classical descriptions of molecular PES. The basic idea underlying the whole derivation is that the experimental fact that the numerous MM models of molecular PES and the VSEPR model of stereochemistry are that successful, as reported in the literature, must have a theoretical explanation [21], The only way to obtain such an explanation is to perform a derivation departing from a certain form of the trial wave function of electrons in a molecule. QM methods employing the trial wave function of the self consistent field (or equivalently Hartree-Fock-Roothaan) approximation can hardly be used to base such a derivation upon, as these methods result in an inherently delocalized and therefore nontransferable description of the molecular electronic structure in terms of canonical MOs. Subsequent a posteriori localization... 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

Such a transformation can be used for relocalizing a given set of delocalized molecular orbitals in conformity with the chemical formula. For instance, the occupied orbitals of methane can be transformed into orbitals very close to simple two-center MO s constructed from tetrahedral sp3 hybrid orbitals and Is hydrogen orbitals 24,25,26) a. unitary transformation can hardly modify the wave function, except for an immaterial phase factor therefore, it leads to a description which is as valid as that in terms of the canonical delocalized Hartree-Fock orbitals. Of course, the localization obtained in this way is not perfect, but it is usually much better than is often believed. In the case of methane, the best localized orbitals are uniquely determined by symmetry 27> for less symmetric molecules one needs a criterion for best localization 28 29>, a problem on which we shall not insist here. A careful inspection reveals that there are three classes of compounds ... 

Veillard is not convinced that ab initio Hartree-Fock theory will ever lead to the kind of black box calculations now possible for some organic molecules. Tsipis is rather more positive recognising the possibilities for a complementary interplay between theory and experiment but is nevertheless of the opinion that there exists no general canon for the efficacious selection and application of the most eligible computational method for the study of a certain compound or series of compounds . This view does not appear to be shared by Ziegler or Comba who convey quite positive messages concerning the capabilities of Density Functional Theory and Molecular Mechanics respectively. 

Density functional theory also offers an attractive computational scheme, the Kohn-Sham (KS) theory [2], similar to the Hartree-Fock (HF) approach, which in principle takes into account both the electron exchange and correlation effects. The canonical KS orbitals thus offer certain interpretative advantages over the widely used HF orbitals, especially for describing the bond dissociation and the open system characteristics, when the electrons are added or removed from the system [3,82,126-130]. For this reason, a determined effort has been made to calculate the reactivity indices from the KS DFT calculations [3,82,83,112,118,119,121, 131-136]. 

In Table 1, we list the initial and optimal expansion coefficients and orbital parameters for Clementi-Roetti-type [89] Is and 2s functions. The optimization procedure was carried out at fixed density p(r) = phf t) (where the Hartree-Fock density is that associated with the Clementi-Roetti wavefunction), taking as variational parameters the expansion coefficients and the orbital exponents. To maintain orbital orthogonality, after each change in the variational parameters, the orbitals were subjected to Schmidt orthogonalization. Notice that the optimal parameters appearing in Table 1 correspond to non-canonical Hartree-Fock orbitals. 

Suppose states D > and A > are one-determinant many-electron functions, which are written in terms of (real) molecular orbitals and where a is the spin index, a a, p. These are the optimized canonical orbitals obtained from Hartree-Fock calculations of states D and A. Using the standard rules of matrix element evaluations[18], one can obtain an appropriate expression for Eq. (1) in terms of MO s of the system. 

When the single determinant many-electron functions are constructed from canonical Hartree-Fock orbitals, the excited functions, and , are doubly excited with respect to the reference function . The second term in the third order energy expression cancels diagonal components for which p = v in the first term. The principal term in the fourth order energy expression has the form... 

Both and can involve double replacements with respect to the single determinantal reference function constructed from canonical Hartree-Fock orbitals. The intermediate state can involve single, double, triple and quadruple replacements with respect to the reference function. The renormalization terms in (44) involve only double replacement. Brueckner91 showed that diagonal terms in the principal term which have a non linear dependence of N are completely cancelled by the renormalization terms. This cancellation is incomplete if the level of replacement employed in generating the intermediate states is restricted. 

Figure 8 Second order correlation energy diagrams for a closed-shell system described in zero order by a single determinantal Hartree-Fock wave function constructed from canonical orbitals. The one Brandow diagram of this type is shown in (a). The exchange diagrams are shown in Goldstone form in (b)...

Within the closed-shell SCF MO formalism one can express the energy as a function of n, E °(n). The MO occupations can now be considered as continuous, independent variables, for the frozen MO shapes [52], in the spirit of the Hyper-Hartree-Fock or X approaches of Slater [60]. The first order variation in energy [59], 8E °(n) = Mobn, identifies /Mmo as equal to the canonical MO energies ... 

The concept of localized MOs is not as widely applicable as that of delocalized canonical MOs. Delocalized MOs are valid for any molecule. However (as noted in Section 11.5), the Hartree-Fock wave functions of nonclosed-shell electronic states are, in most cases, linear combinations of several Slater determinants [for example, see (10.44) and (10.45)], and the above localization procedure does not apply to the open-shell orbitals in these wave functions. Thus, in a molecule in an excited electronic state with an open-shell configuration, the electrons in the incompletely filled MOs are delocalized over much of the molecule. 

Section 3.2 constitutes a derivation of the results of the previous section. The order of presentation of these two sections is such that the derivations of Section 3.2 can be skipped if necessary. For a fuller appreciation of Hartree-Fock theory, however, it is recommended that the derivations be followed. We first present the elements of functional variation and then use this technique to minimize the energy of a single Slater determinant. A unitary transformation of the spin orbitals then leads to the canonical Hartree-Fock equations. 

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