Wave zero order

If the solution of the zero-order Schiodinger equation [i.e., all teiins in (17) except V(r,Ro) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling tenns. If F) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

These are zero-, first-, second-, th-order perturbation equations. The zero-order equation is just the Schodinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy, Wi, and the first-order correction to the wave function, 4< i. The th-order energy correction can be calculated by multiplying from the left by 4>o and Integrating, and using the turnover rule ( o Ho, ) = (, Ho o)... 

The zero-order wave function is the HF determinant, and the zero-order energy is just a sum of MO energies. The first-order energy correction is the average of the perturbation operator over the zero-order wave function (eq. (4.36)). 

The main limitation of perturbation methods is the assumption that the zero-order wave function is a reasonable approximation to the real wave function, i.e. the perturbation operator is sufficiently small . The poorer the HF wave function describes... 

The zero-order ground state spin-orbit wave functions for these systems, obtained as previously described (41), have therefore been used to calculate the magnetic susceptibilities via the Van Vleck equation... 

In the first approach it is assumed, as well, that the reaction proceeds by zero-order. Since the rate term d> is not a function of concentration, the continuity equation is not required so we can deal with the more convenient energy equation. Semenov, like Mallard and Le Chatelier, examined the thermal wave as if it were made up of two parts. The unbumed gas part is a zone of no chemical reaction, and the reaction part is the zone in which the reaction and diffusion terms dominate and the convective term can be ignored. Thus, in the first zone (I), the energy equation reduces to... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Thus, the first-order correction to the zero-order (unperturbed) wave function is obtained by substituting equation (A.104) into equation (A.103) and changing the summation index ... 

The diagonal term in = n is excluded from the summation in equation (A. 105) since that wave function is the zero-order term. The summation should converge at some finite value of in as the energy difference in the denominator becomes large. 

Moller-Plesset perturbation theory (MPPT) aims to recover the correlation error incurred in Ilartree- Fock theory for the ground state whose zero-order description is ,. The Moller-Plesset zero-order Hamiltonian is the sum of Fock operators, and the zero-order wave functions are determinantal wave functions constructed from HF MOs. Thus the zero-order energies are simply the appropriate sums of MO energies. The perturbation is defined as the difference between the sum of Fock operators and the exact Hamiltonian ... 

A0 + 2 A cos 2n(nX—an), where each term represents the contribution of one order, the coefficients Av A2, etc., being the amplitudes of the waves (the square roots of the intensities of the diffraction spots), and ot19 a2, etc., the phase displacements. (A0 is the amplitude of the zero-order diffraction.) For a centro-symmetric pattern, such as the one we are considering, the phase displacements are all either Oor J (in degrees, 0° or 180°) with respect to the centre of symmetry, and therefore the expression... 

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