Molecular orbital theory variational computation

It is important to realize that whenever qualitative or frontier molecular orbital theory is invoked, the description is within the orbital (Hartree-Fock or Density Functional) model for the electronic wave function. In other words, rationalizing a trend in computational results by qualitative MO theory is only valid if the effect is present at the HF or DFT level. If the majority of the variation is due to electron correlation, an explanation in terms of interacting orbitals is not appropriate. 

In order to make a correct analysis of such an experimental spectrum, an appropriate theoretical calculation is indispensable. For this purpose, some of calculational methods based on the molecular orbital theory and band structure theory have been applied. Usually, the calculation is performed for the ground electronic state. However, such calculation sometimes leads to an incorrect result, because the spectrum corresponds to a transition process among the electronic states, and inevitably involves the effects due to the electronic excitation and creation of electronic hole at the core or/and valence levels. Discrete variational(DV) Xa molecular orbital (MO) method which utilizes flexible numerical atomic orbitals for the basis functions has several advantages to simulate the electronic transition processes. In the present paper, some details of the computational procedure of the self-consistent-field (SCF) DV-Xa method is firstly described. Applications of the DV-Xa method to the theoretical analysises of XPS, XES, XANES and ELNES spectra are... 

Molecular orbital theory as applied to the hydrogen molecule ion is described fully in Computer Lab Vll. In the solution block below the variational expression for the energy is given along with the first derivative of the ene with respect to the variation parameters alpha and R. These three equations are solved to give the ground-state energy and the optimum values of alpha and R. 

How does this help us The best possible orbital will therefore be the one with the minimum energy. In modem molecular orbital theory, computer programs are designed to try many different variations of a gnessed orbital and compare the energies of each one. The variation with the lowest energy is the best approximation for the actual molecular orbital. 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

To assign values to the molecular orbital coefficients, c, many computational methods apply Hartree-Fock theory (which is based on the variational method).44 This uses the result that the calculated energy of a system with an approximate, normalized, antisymmetric wavefunction will be higher than the exact energy, so to obtain the optimal wavefunction (of the single determinant type), the coefficients c should be chosen such that they minimize the energy E, i.e., dEldc = 0. This leads to a set of equations to be solved for cMi known as the Roothaan-Hall equations. For the closed shell case, the equations are... 

Semiempirical molecular orbital (SEMO) methods have been used widely in computational studies [1,2]. Various reviews [3-6] describe the underlying theory, the different variations of SEMO methods, and their numerical results. Semiempirical approaches normally originate within the same conceptual framework as ab initio methods, but they overlook minor integrals to increase the speed of the calculations. The mistakes arising from them are compensated by empirical parameters that are introduced into the outstanding integrals and standardized against reliable experimental or theoretical reference data. This approach is successful if the semiempirical model keeps the essential physics and chemistry that describe the behavior of the process. 

Yet, even the hnite dimensional standard VB approach runs into a number of difficulties, such as the nightmare of the inner shells [11], neglect of overlap integrals, and the so-called V catastrophe (see, e.g. Ref. [12]). For this very reason, sometime during the second World War, VB theory was eclipsed by the computationally much more amenable molecular orbital (MO) method, relying on the independent-particle model (IPM), which reduces the V-electron problem to effectively a one-electron, though highly non-linear, one. A very important conceptual advance was achieved by the exploitation of the variation principle, which led to the formulation of Hartree-Fock (HF) equations... 

How can we extend HF theory to incorporate the effects of the most important natural orbitals, even in cases where the occupation numbers are not close to two or zero Actually, Lowdin gave an answer to this question in his 1955 article, where he derived something he called the extended HF equations. The idea was to use the full Cl wave fimction. Equation (2), but with a reduced number of orbitals, and determine the expansion coefficients and the molecular orbitals variationally. His derivation was formal only and had no impact on the general development at the time. It was not xmtil 20 years later that a similar idea was suggested and developed into a practical computational procedure. The approach is today known as the complete active space SCF method, CASSCF. ... 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

Ab initio modem valence bond theory, in its spin-coupled valence bond (SCVB) form, has proved very successful for accurate computations on ground and excited states of molecular systems. The compactness of the resulting wavefunctions allows direct and clear interpretation of correlated electronic structure. We concentrate in the present account on recent developments, typically involving the optimization of virtual orbitals via an approximate energy expression. These virtuals lead to higher accuracy for the final variational wavefunctions, but with even more compact functions. Particular attention is paid here to applications of the methodology to studies of intermolecular forces. 

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