2n + I rule

Table 1 The Orders of the Responses Needed to Calculate the Energy to a Given Order in the Perturbation According to the 2n + I Rule for the Wavefunction Parameters and the 2n+2 rule for the Lagrange Multipliers...

From this it would appear that the ( - l)th-order wave function is required for calculating the th-order energy. However, by using the turnover rule and the nth and lower-order perturbation equations (4.32), it can be shown that knowledge of the nth-order wave function actually allows a calculation of the (2n-i-l)th-order energy. 

For variationally optimized wave functions (HF or MCSCF) there is a 2n -I- 1 rule, analogous to the perturbational energy expression in Section 4.8 (eq. (4.34)) knowledge of the Hth derivative (also called the response) of the wave function is sufficient for... 

The Lagrange technique may be generalized to other types of non-variational wave functions (MP and CC), and to higher-order derivatives. It is found that the 2n - - 1 rule is recovered, i.e. if the wave function response is known to order n, the (2n + l)th-order property may be calculated for any type of wave function. 

Balakrishnarajan and Jemmis [32, 33] have very recently extended the Wade-Mingos rules from isolated borane deltahedra to fused borane ("conjuncto ) delta -hedra. They arrive at the requirement of n I m skeletal electron pairs corresponding to 2n + 2m skeletal electrons for such fused deltahedra having n total vertices and m individual deltahedra. Note that for a single deltahedron (i.e., m = 1) the Jemmis 2n + 2m rule reduces to the Wade-Mingos 2n I 2 rule. 

Therefore, m solutions of linear equations (with a perturbation-dependent vector like W on the right-hand side) can replace the 0 m ) solutions for Note that the situation, though similar, is not completely analogous to the case of the (2n -I- 1) rule. 

Having evaluated R i, the corresponding number of plates N can be determined. The calculation is truncated by employing another useful rule of thumb that when R = 2R the number of plates required in the column will be JV = 2N i , where iV is the number of plates required at total reflux. Determination of N begins with Fenske s relationship ... 

Nickel and carbon monoxide form a variety of carbonyl compounds with the general formula Nim(CO) . Derive the structure of the following compounds by employing Wades 2n + 2 rules and determine the symmetry point group of each of your structures, (i) Nis(CO)ii (ii) Ni6(CO)i4 (iii) Ni7(CO)i7 (iv) Ni8(CO)i8 (v) Ni9(CO)i9... 

Table 3.1 Rules for the association of dynamic variables to operators. The square of a variable corresponds to double application of the operator times means that the variable transforms into a multiplier, h is Planck s constant, italic h = h/2n i =. /— 1, i = —1. mp is the mass of the particle...

We have established that for a fully variational wavefunction we may calculate the molecular gradient from the zero-order response of the wavefunction (i.e., from the unperturbed wavefunction) and the molecular Hessian from the first-order response of the wavefunction. In general, the 2n + 1 rule is obeyed for fully variational wavefunctions, the derivatives (responses) of the wavefunction to order n determine the derivatives of the energy to order 2n -i- 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wavefunction to first order, but that the calculation of the energy to fourth order requires a knowledge of the wavefunction response to second order. These relationships are illustrated in Table 1. 

Having seen how the evaluation of energy derivatives for fully variational wavefunctions is simplified by the 2n -i- 1 rule, let us now consider the molecular gradient for the simplest model of ab initio theory - the self-consistent field (SCF) model. We will see that, although the result for fully variational wavefunctions (equation 18) may be applied also to SCF energies, we cannot do this without first considering carefully the functional form of the energy expression. With minor modifications, the formalism presented here may be applied also to density-functional theory (DFT). 

For variational wavefunctions, the wavefunction parameters obey the 2ri + I rule, which states that the derivatives of the parameters to order n determine the derivatives of the energy to order 2n + I. The same rule is obeyed also by the wavefunction parameters of nonvariational wavefunctions, whereas the Lagrange multipliers follow the 2n +2 rule. 

Combining the results for perturbations of even and odd orders, we conclude that the wave-function parameters to order n determine the RSPT Lagrangian to order 2n + 1 (Wigner s 2n + I nde) [1] and that the associated multipliers to order n determine the Lagrangian to order 2n -1-2 (the 2n -I- 2 rule) [2]. 

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