Zeroth order equations

The zeroth order equations ean easily be solved beeause HO is independent of time. Assuming that at t = - T = /i (we use the index i to denote the initial state), this solution is ... 

If the coefficients Ckk(Q) and c xiQ) and the operators A x are sufficiently small, the summation on the left-hand side of equation (10.18) and Ckk(Q) in (10.19) may be neglected, giving a zeroth-order equation for the nuclear motion... 

Zeroth-order reactions are those in which the rate is independent of the concentration of any reactants. The zeroth-order equation is... 

Term. The zeroth order equation is simply c = c, a constant to be determined by matching. ... 

Isothermal nth Order Irreversible Reaction. Here f = c and the zeroth order equation representing the outer solution is... 

It is important to stress that the leading term of the RS perturbation Equations (4.7), the zeroth- order equation (Ho—Eo) Ao = 0, must be satisfied exactly, otherwise uncontrollable errors will affect the whole chain of equations. Furthermore, it must be observed that only energy in first order gives an upper bound to the true energy of the ground state, so that the energy in second order, E(2), may be below the true value. 

If only one pair of measurements had been made, say (> i,Xi), then a zeroth order equation of the type = yi, for all> would be the only possible solution. With two pairs of measurements, ( i,Xi) and (y2,X d then a first-order linear model can be proposed. 

The starting point for direct perturbation theory, to be discussed in detail in chapter 12, is equation (123) with the L6vy-Leblond equation as the zeroth-order equation. Solving for the small components we obtain the following equation... 

In order to derive a useful perturbation theory expression with equation (21) as the zeroth-order equation, the modified Dirac equation (20) has to be reformulated in such a way that the operator difference between equations (20) and (21) can be identified and used as a perturbation operator. 

In such a theory, the zeroth-order equation of motion for the system coordinate is... 

These are the zero-, first-, second-, nth-order perturbation equations. The zeroth-order equation is just the Schrddinger 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, Fi. The nth-order energy correction can be calculated by multiplying from the left by 4>o and integrating, and using the "turnover mfe o>. ... 

If the expressions for ff, and E from equations (1) - (3) are used in equation (4), and terms with the same powers of A are collected, one obtains a set of equations, one for each power of A. The zeroth order equation is... 

Ad (iii) Accepting the partition described by Eq. (13.12), the solution of the zeroth-order equation involves the solution of the Hartree-Fock problem. We have to specify the ground states and excited many-electron states explicitly. The ground state is simply the Fermi vacuum ... 

Since the zeroth-order equations are solved only in the Hartree-Fock approximation, the perturbation corrections account not only for intermolec-ular interactions, but for the intramolecular correlation energy as well. These two effects cannot be separated in the Lowdin basis set, but one may subtract the contributions of those [ij kl] integrals which result in local correlation. 

In the previous two sections, we have presented the Breit-Pauli perturbation Hamiltonian for one- and two-electron relativistic corrections of order 1/c to the nonrelativistic Hamiltonian. But there is a problem for many-electron systems. For the perturbation theory to be valid, the reference wave function must be an eigenfunction of the zeroth-order Hamiltonian. If we take this to be the nonrelativistic Hamiltonian and the perturbation parameter to be 1/c, we do not have the exact solutions of the zeroth-order equation. 

The zeroth-order equation has been discussed above. The kinetic balance relation holds between the large and small components of the zeroth-order wave function. Intermediate normalization has been chosen for the zeroth-order wave function. 

The variation equation for the second-order wave function comes from the fourth-order energy. If we were to make the variation of the second-order wave function in the second-order energy expression we would only get the zeroth-order equation again, so we need a functional that is quadratic in the second-order wave function. Variation of the fourth-order energy subject to the normalization conditions gives the equation... 

The second equation is the same as for the zeroth-order equation, and we immediately arrive at the nonrelativistic equation by substituting the second into the first,... 

Now, making a perturbation series in with the physical value /x = 1/c, the zeroth-order equation is... 

To improve on the lORA energies, a perturbation series may be developed based on the lORA equation. This is most simply done by comparing with the perturbation series for ZORA, (18.22). As we mentioned above, lORA corresponds to putting the term that is linear in the energy into the metric to make it part of the zeroth-order equation. Because of this, the first-order energy and the first-order wave function are zero. The second-order energy is... 

The two sides of this equation are power series in A. Therefore, the terms multiplied by the same powers of A, i.e. terms of the same order, have to be equal on both sides in order for the whole equation to be fulfilled. We thus obtain a series of equations, where the zeroth-order equation is just the equation for the unperturbed Hamiltonian, Elq. (3.14) again. The first- and second-order equations are... 

Each term in this expansion obeys the equations which are obtained by substituting (3.125) and (3.129) into Eq. (3.128) and collecting the terms of the same order. It is easy to see that the zeroth-order equation ... 

So far, the possibility of optimizing the orbitals in the presence of a perturbation (i.e. of making self-consistent property calculations) has been considered only at the Hartree-Fock level. In many cases, however, it is necessary to use a many-determinant wavefunction, either because the IPM ground state is degenerate or because electron-correlation effects are too important to be ignored and it is then desirable to optimize both Cl coefficients and orbitals as in MC SCF theory (Section 8.6). To formulate the perturbation equations, both coefficients and orbitals will be expanded in terms of a perturbation parameter and the orders will be separated the zeroth-order equations will be the MC SCF equations in the absence of the perturbation, while the first-order equations will determine the (optimized) response of the wavefunction, and will thus permit the calculation of second-order properties. Important progress had been made in this area (Jaszunski, 1978 Daborn and Handy, 1983), for particular types of perturbation and Cl function. In fact, however, the equations in their most general form have been known for many years (Moccia, 1974), and are implicit in the stationary-value... 

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