Molecular orbitals minimizing energy with respect

Efforts to deduce transition state structures theoretically have until recently been retarded by the failure of even the more sophisticated molecular orbital treatments to predict accurate activation energies, and the need to avoid geometric and mechanistic assumptions has made the calculation of reaction pathways prohibitively expensive. The introduction of efficient gradient methods for minimizing energy with respect to all geometric parameters, coupled with the advent of faster computers, has now virtually overcome the latter problem, and careful parameterization of semiempirical molecular orbital methods has led to more... 

The Hartree-Fock approximation obtained by using a single determinantal wavefunction, eq 4, constructed from molecular orbitals given by eq 6, in the expression for the variational energy, eq 3, and minimizing the energy with respect to the molecular orbital coefficients, C i. 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

In any molecule, the potential close to a nucleus approximates that of the free atom (in the correct valence state) consequently, in these regions the molecular orbitals lie near the free atomic orbitals. Accordingly, it is generally assumed that the molecular orbitals may be expressed as linear combinations of the separate atomic orbitals, the LCAO approximation. The total electronic energy of the molecule may then be minimized with respect to the coefficients in this approximation and the form of the molecular orbitals thus determined. If the molecular orbitals 0, are expressed in terms of the set of atomic orbitals < b. .., N as... 

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) ... 

---