Molecular orbital theory wave-function coefficients

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

This separation of the cr framework and the re bond is the essence of Hiickel theory. Because the re bond in ethylene in this treatment is self-contained, we may treat the electrons in it in the same way as we do for the fundamental quantum mechanical picture of an electron in a box. We look at each molecular wave function as one of a series of sine waves, with the limits of the box one bond length out from the atoms at the end of the conjugated system, and then inscribe sine waves so that a node always comes at the edge of the box. With two orbitals to consider for the re bond of ethylene, we only need the 180° sine curve for re and the 360° sine curve for re. These curves can be inscribed over the orbitals as they are on the left of Fig. 1.23, and we can see on the right how the vertical lines above and below the atoms duplicate the pattern of the coefficients, with both c and c2 positive in the re orbital, and c positive and c2 negative in re. ... 

At the HF-SCF level of theory, the wave function is determined completely by the molecular orbitals. In the vast majority of cases, these are given by linear combinations of atom-centered basis functions, and these MO coefficients are obtained by the self-consistent field procedure. The first-order change to the wave function is therefore governed by the first-order change in the MO coefficients. It is not difficult to work out expressions for the derivatives of the MO coefficients,22 and one obtains... 

Moreover, the second-generation MCSCF parametrizes the wave function in a way that enables the simultaneous optimization of spinors and Cl coefficients, in this context then called orbital or spinor rotation parameters and state transfer parameters, respectively. Then, a Newton-Raphson optimization method is employed which also requires the second derivatives of the MCSCF electronic energy with respect to the molecular spinor coefficients (more precisely, to the orbital rotation parameters) and to the Cl coefficients. As we have seen, in Hartree-Fock theory the second derivatives are usually not calculated to confirm that a solution of the SCF procedure has indeed reached a minimum with respect to the large component and not a saddle point. Now, these general MCSCF methods could, in principle, provide such information, although it is often not needed in practice. 

A model of carbon bonding established to account for the tetrava-lent nature of carbon involves two steps, in which the atoms are first raised to excited states (valence states), and are then permitted to interact in these states to form the molecule. In the case of carbon, one of the 2s orbitals is thus "promoted" to a p orbital to make available four unpaired electrons. The result is the "hybridization" of an s orbital with three p orbitals to form four equivalent and tetrahedrally oriented sp orbitals. To devise molecular orbitals, one takes a linear combination of atomic orbitals. Following the valence bond theory, the atoms are in their excited state when hybridized, and then come together to form a molecule. In the molecular orbital approach, when the proper coefficients for the wave functions of the linear combination of atomic orbitals are chosen—i.e., those coefficients which will minimize the energy of the resultant molecule—the results will be the same as when hybrid orbitals are employed. Thus, both approaches lead to a minimization of energy and to stable bond formation. 

Car and Parrinello in their celebrated 1985 paper [2] proposed an alternative route for molecular simulations of electrons and nuclei altogether, in the framework of density functional theory. Their idea was to reintroduce the expansion coefficients Cj(G) of the Kohn-Sham orbitals in the plane wave basis set, with respect to which the Kohn-Sham energy functional should be minimized, as degrees of freedom of the system. They then proposed an extended Car-Parrinello Lagrangian for the system, which has dependance on the fictitious degrees of freedom Cj(G) and their time derivative Cj (G) ... 

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