2n+l rule

We have remarked that there is a large number of coupled HF/KS equar tions to be solved in order to get electric response properties. However, as for the case of a molecule in vacuo, also for solutes the computational effort can be reduced by resorting the exploiting to so called (2n+l) rule that permits to get the (.2n+l)th response using nth derivative of P. ... 

Let us assume the intermediate Hamiltonian of the DKH procedure after n transformation steps were given. From the (2n- -l)-rule [611,634] it may be written as... 

In accordance with the (2n + l)-rule the application of Hi (A) yields the final second- and third-order even terms, whose most compact form seems to be identical to the familiar unperturbed expressions. 

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. 

As in perturbation theory, there is a 2n-l-l rule [45 7], Using the nth-order wavefunction, energy derivatives up to 2n+l may be evaluated directly, and the derivative wavefunctions between the n and 2n -F1 orders are not required explicitly. For example, with the first-derivative wavefunctions known explicitly, the third energy derivatives may be calculated directly. To see this for the abc derivative of Eqn. (38), one evaluates the equation at the equilibrium choice of parameters and then integrates with the zero-order wavefunction. That is,... 

The equations of quantum mechanics and rules for their application are needed for deeper purposes than the mere prescription of energy levels. The wave equation itself is much more than a vehicle for the values of E which permit solutions the properties of the function itself assume great importance. One example has already appeared in connexion with rotational states for the th state there are 2n- -l solutions and this number determines the weight of the state. 

As an example of the analytical derivatives of the PCM energy functional, which can be formulated in agreement with a (2n- -l) Wigner perturbation rule [1, 2], we will consider the analytical derivatives for the coupled-cluster level up to third order of differentiation. 

Table 14.1 Wigner s 2n + 1 rule for the wave-function parameters and the 2n -l- 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. 

The interactions of type (1) are known as L-S coupling or Russell-Saunders coupling. From the vector addition rule and the constraint that the values must differ by one in the unit of h/2n, the possible values of L and S are ... 

The general VLE problem treats a multicomponent system of N constituent species for which the independent variables are T, P, N—1 liquid-phase mole fractions, and N —1 vapor-phase mole fractions. (Note mat = 1 and L yi = 1, where Xi and represent liquid and vapor mole fractions, respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations... 

Fig. 1. Schematic of an RX pattern for a 1D doubling of unit cell by translation of one atom over two by p the symmetry breaking induced by the phase transition modifies the extinction rules, and the Debye-Waller factors. Only the diffraction by the mean unit cell is considered, not diffuse scattering. Left the Bragg peaks at qn = n2n/a are modulated by the Debye-Waller factor with mean square fluctuations < Su2y0. Right by moving one atom over two by p in a given direction, the unit cell is doubled and superstructures at qm = (2ra+ l)2n/2a appear, with amplitude (qp)2 when qp< 1.

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