Intermediate normalized wave function

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

Perturbation methods add all types of corrections (S, D, T, Q etc.) to the reference wave function to a given order (2, 3, 4 etc.). The idea in Coupled Cluster (CC) methods is to include all corrections of a given type to infinite order. The (intermediate normalized) coupled cluster wave function is written as... 

For the closed shell ground state, Sp includes ( )o. Furthermore, for the same state, but in the MR-SDCI cases, other c ) are included in Sp, but their weight in the wave function will always remain lower than unity. Hence, we can write, in the intermediate normalization... 

The C-conditions ascertain the validity of the intermediate normalization, Eq. (6), by requiring that the off-diagonal transformed coefficients c that are associated with the reference configurations ] j), (j i) in the target wave function must vanish, since... 

The addition of two angular momenta (formula of the type (10.4)) may be directly generalized to cover the case of an arbitrary number of momenta. However, in such a case it is not enough to adopt the total momentum and its projection for the complete characterization of the wave function of coupled momenta. Normally, the quantum numbers of intermediate momenta must be exploited too. Moreover, these functions depend on the form (order) of the coupling between these momenta. The relationships between the functions, belonging to different forms of coupling of their momenta, may be found with the aid of transformation matrices. 

In this section we will investigate the nature of the exact perturbed wave function through Rayleigh-Schrodinger perturbation theory. We will employ intermediate normalization... 

Both VU and SU MR CC methods employ the effective Hamiltonian formalism the relevant cluster amplimdes are obtained by solving Bloch equations and the (in principle exact) energies result as eigenvalues of a non-Hermitian effective Hamiltonian that is defined on a finite-dimensional model space Mq. An essential feature characterizing this formalism is the so-called intermediate or Bloch normalization of the projected target space wave functions I f, ) with respect to the corresponding model space configurations 1, ), namely = 8 (for details, see, e.g. Refs. [172,174]). 

Exploring the cluster analysis of a finite set of FCI wave functions based on the SU CC Ansatz [224], we realized that by introducing the so-called C-conditions ( C implying either constraint or connectivity , as will be seen shortly), we can achieve a unique representation of a chosen finite subset of the exact FCI wave functions, while preserving the intermediate normalization. (In fact, any set of MR Cl wave functions can be so represented and thus reproduced via an MR CC formalism.) These C-conditions simply require that the internal amplitudes (i.e. those associated with the excitations within the chosen GMS) be set equal to the product of aU lower-order cluster amplitudes, as implied by the relationship between the Cl and CC amplitudes [223], rather than by setting them equal to zero, as was done in earlier IMS-based approaches [205,206] (see also Ref. [225]). Remarkably, these conditions also warrant that all disconnected contributions, in both the elfective Hamiltonian and the coupling coefficients, cancel out, leaving only connected terms [202,223]. 

An important insight in the development of size-extensive formulations in a IMS was the realization that the intermediate normalization convention for the wave operator, viz. PflP = P, should be abandoned in favor of a more appropriate normalization [28,61]. For the IMS, in general, products of quasi-open operators may lead to internal excitations, or may even be closed, so that if we choose il = 2 exp(7 )l< X< l, with = Top-f Tq-op, then powers of Tq. op coming from the exponential might lead from 4>p to internal excitations to some other model function or it may contain closed operators. We would have to bear this in mind while developing our formalism, and would not force POP = P in our developments. 

Perturbative solution of the Bloch equation. We will first introduce the reduced wave operator X by assuming intermediate normalization for the exact wave function and writing fl as... 

As mentioned in the introduction, the above discussion of the small-, large-, and intermediate-molecule limits of electronic relaxation processes can also be utilized with very minor modifications to discuss the phenomena of intramolecular vibrational relaxation in isolated polyatomic mole-cules. ° Figure 4 is still applicable to this situation. The basis functions are now taken to be either pure harmonic vibrational states, some local-mode vibrational eigenfunctions, or some alternative nonlinear mode-type wave-functions. In the following the nomenclature of vibrational modes is utilized, but its interpretation as normal or local can be chosen to suit the circumstances at hand. 

The size-consistency problem has been removed by using the coupled cluster (CC) ansatz [12, 118, 119, 120, 121, 122,123,124, 125). While in Cl one writes the wave function as (in intermediate normalization)... 

In intermediate normalization, the projector P and the wave operator 2 can be understood (and illustrated) in a very simple way While P projects an arbitrary state - a) e H onto the model space, the wave operator 2 transforms each function with the projection 1 ) M back onto the exact state ). In intermediate normalization, this property is independent of the (size of the) component which lies within the complementary space H M), owing to the two properties P = Q and QQ = 0, respectively. 

If i/f does not satisfy this equation, then multiplication of by the constant gives a perturbed wave function with the desired property. The condition called intermediate normalization, simplifies the derivation. Note that multiplication of i/f by a constant does not change the energy in the Schrbdinger equation H = E , so use of intermediate normalization does not affect the results for the energy corrections. If desired, at the end of the calculation, the intermediate-normalized can be multiplied by a constant to normalize it in the usual sense. 

It is convenient to choose the perturbed wave function to be intermediately normalized, i.e. the overlap with the unperturbed wave function should be 1. This has the consequence that all correction terms are orthogonal to the reference wave function. 

Using intermediate normalization, a Cl wave function can be generated by allowing the excitation operator to work on an HF wave function. 

The coefficients in the CISDTQ and CCSDTQ wave functions (using intermediate normalization) for the dominating excitations are given in Table 4.6. 

The problem, however, lies in the fact that these doubly excited determinants are equipped with coefficients obtained in the full Cl method (i.e., with all possible excitations). How is this We should draw attention to the fact that, in deriving the formula for A , intermediate normalization is used. If someone gave us the normalized FCl wave functions as a Christmas gift, then the coefficients occurring in the formula for AE would not be the double excitation coefficients in the FCI function. We would have to denormalize this function to have the coefficient for the Hartree-Fock determinant equal to 1. We cannot do this without knowledge of the coefficients for higher excitations. 

An intermediate normalization of the wave function the normalized function means that ... 

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