Molecules total electronic energy

Using this expression for the total electronic energy, application of the variational principle yields the following set of differential equations to obtain the optimized spatial MOS, 4, for the molecule ... 

However, in a large number of closed shell molecules, a single Slater determinant describes the ground state wave function fairly accurately. Even in such cases inclusion of excited state configuration results in substantial lowering of total electronic energy, and this is referred to as nondynamic electron correlation. 

When a molecule A is attacked by another molecule B, it will be perturbed in either its number of electrons NA or its external potential vA(r). At the very early stages of the reaction, the total electronic energy of A, EA can be expressed as a Taylor series expansion around the isolated system values NA and v jfr)... 

Nevertheless, in spite of all these efforts, it has proved exceedingly difficult to compute the necessary quantities (i.e., total electronic energies) to a target accuracy of better than 1 kj/mol, which is the typical accuracy of experimental measurements of heats of formation. In this chapter, we examine the ab initio calculation of atomization energies (AEs) of gas-phase molecules, from which the heats of gas-phase reactions between the same molecules can be easily obtained. Our purpose is not only to illustrate the inherent difficulties associated with the accurate calculation of AEs, but also to describe the considerable progress that has been achieved over the last few years and the perspectives for the near future. 

Two key pieces of theoretical data are required to obtain an empirical estimate of the correlation energy from the experimental data collected in the preceding section the total molecular Hartree-Fock energy and the relativistic corrections . It is implicit in the definition of the correlation energy presented in Eq. (1) that the total electronic energy ofa given molecule, Ef, may be divided into three constituent parts,... 

FIGURE 19. B3LYP/6-31 H-G(d,p)-optimized transition stmctures for the epoxidation of cyclohexene with DMDO in the presence of one (a) and two (h) water molecules. The transition structure for the epoxidation of -2-butene (c) is optimized at the same level of theory in the presence of one water molecule. The classical barriers are estimated using total electronic energies of the transition stmctures, cyclohexene (—234.71316 au), ii-2-butene (—157.27453 au), DMDO with one water molecule (—344.81523 au) and DMDO with two water molecules (—421.28672 au)... 

In the Fock operator, the core Hamiltonian h( 1) does not depend on the orbitals, but the Coulomb and exchange operators (1) and ( 1) depend on ( 1). If (1,2,3,..., Ne) is constructed from the lowest energy Ne orbitals, one has the lowest possible total electronic energy. By Koopmans theorem, the negative of the orbital energy is equal to one of the ionization potentials of the molecule or atom. 

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... 

T(l,2,...n) is the electronic wavefunction and explicitly is a function of the coordinates of all n electrons in this notation the coordinates of a given electron are symbolized by a single number. E is the total electronic energy of the molecule. 

Jahn-Teller effects,13 which are the chemist s analog of electron-phonon coupling mechanisms in molecules, are consequences of the application of perturbation theory. With respect to some distortion coordinate si the total electronic energy for a system may be expressed to second order as... 

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

The SHM permits the calculation of a kind of stabilizing energy, or, more accurately, an energy that reflects the stability of molecules. This energy is calculated by comparing the total electronic energy of the molecule in question with that of a reference compound, as shown below for the propenyl systems, cyclobutadiene, and the cyclobutadiene dication. 

The propenyl cation, Fig. 4.19b cf. Fig. 4.16. If we take the total % electronic energy of a molecule to be simply the number of electrons in a n MO times the energy level of the orbital, summed over occupied orbitals (a gross approximation, as it ignores interelectronic repulsion), then for the propenyl cation... 

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