The Perturbed Wavefunction

Having now carried out some detailed analysis of the RSPT expression for Ej, let us turn to the perturbative corrections to the wavefunction y . The first-order RSPT wavefunction is, according to Eq. (3.9), 

As we did above for the energy, it is instructive to analyze when it pertains to two noninteracting subsystems (a and b). For this special case, the sum in Eq. (3.46) separates into terms pertaining to each of the isolated systems  

we see that, through first order, the wavefunction jy contains only terms of the form and terms such as are not 

It is natural to ask how the RSPT can have the physically consistent property that its energy is extensive whereas its wavefunction does not reduce to a product form for noninteracting systems. The answer has to do with the manner in which the total energy Ej is computed in perturbation theory  

In contrast, the total energy is obtained, in variational approaches, by eval- 

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The assumption that V is a small perturbation to Hg suggests that the perturbed wavefunction and energy can be expressed as a power series in V. The usual way to do so is in terms of the parameter X ... 

The perturbed wavefunction and energy are substituted back into the Schrodinger equation ... 

If we used perturbation theory to estimate the expansion coefficients c etc., then all the singly excited coefficients would be zero by Brillouin s theorem. This led authors to make statements that HF calculations of primary properties are correct to second order of perturbation theory , because substitution of the perturbed wavefunction into... 

The presence of the second proton induces a perturbation to the wavefunction and the energy eigenvalue of the electron. For the perturbed wavefunction, we make the Ansatz... 

A fruitful approach to the problem of intermolecular interaction is perturbation theory. The wavefunctions of the two separate interacting molecules are perturbed when the overlap is nonzero, and standard treatment [49] yields separate contributions to the interaction energy, namely the Coulombic, polarization, dispersion, and repulsion terms. Basis-set superposition is no longer a problem because these energies are calculated directly from the perturbed wavefunction and not by difference between dimers and monomers. The separation into intuitive contributions is a special bonus, because these terms can be correlated with intuitive molecular... 

Another approach to the evaluation of higher-order energies is to obtain the perturbed wavefunction, or some approximation to it, explicitly. This method has... 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

The ground-state Ln0oX0) wavefunction is utilized as the zeroth order basis set (using rounded kets), and using the singly- and doubly-excited state functions, the perturbed wavefunctions of the LnX, 3 system are written to first order in H. Then the EDV transition moment for the Ln0oX0) —> l /iXo) transition is given by... 

Eor second-order properties derived from the perturbation, we have a particularly attractive method that uses the EOM eigenvectors to represent the perturbed wavefunctions. Since the right- and left-hand eigenvectors form a complete set, we know we can write the perturbed wavefunctions in the form. 

Xip, can be obtained. Without loss of generality, the perturbed wavefunctions can be taken to be orthogonal to the reference function, that is... 

The change in density at atom I equals the square of the perturbed wavefunction minus that of the unperturbed wavefunction and involves only the doubly occupied MOs —m ... 

We note that the atomic axial tensor cannot be expressed directly as an energy derivative since it involves the overlap over two perturbed wavefunctions. However, the perturbed wavefunctions can be obtained as the dot product of the solution vectors N defined by the linear response function in Eq. 2.26 for a magnetic field and nuclear displacement perturbation, respectively. It is important here to note that some care needs to be exercised in the way the orbital rotations, in the language of Section 2.1.1, are defined [253, 254]. 

Finally, we assmne that the eigenfunction o(. )) nd eigenvalue Eo iF) of the full Hamiltonian H are close to those of the unperturbed Hamiltonian i.e. the perturbation by T is indeed small. We can then expand the perturbed wavefunction... 

In order to make the derivations mathematically easier without changing the final expressions we require the perturbed wavefunction o(- )) to be normalized in the following way, called intermediate normalization... 

We are ready now to insert the power-series expansions of the perturbed wavefunction and energy in the Schrodinger equation, Eq. (3.1),... 

In Part III we will apply these expressions directly, but here we want to combine them with perturbation theory as developed in Section 3.2. Inserting thus the perturbation theory expansion of the perturbed wavefunction o(- )) Eq- (316), in the right-hand side of Eq. (3.40) we obtain for the second derivative of the energy... 

As the time dependence of the unperturbed wavefunctions is simply a rotation in the complex plane, Eq. (2.6), we can say that in the interaction picture this rotation is frozen out or that we have switched to a rotating frame that rotates with the time dependence of the unperturbed wavefunctions. The time evolution of the perturbed wavefunction 4 o( )) i the interaction picture thus governed by the perturbation Hamiltonian alone, as we will see in the following section. This will greatly... 

The compact integral expressions for the time-dependent wavefunction in Eq. (3.86) and in particular for the first-order correction in Eq. (3.87) will be employed in the derivation of response functions in the following section. However, for the interpretation of the time-dependent wavefunctions it is useful to expand them in the complete set of unperturbed wavefunctions Eq. (2.14) analogous to the perturbed wavefunctions of time-independent perturbation theory in Eq. (3.23)... 

The first contribution is an expectation value of the second-order perturbation operator with the unperturbed wavefunction of the system. It is negative and is called the diamagnetic contribution The second contribution involves either the first derivative of the perturbed wavefunction, the first-order correction to the wavefunction, a sum over all the other unperturbed states or a linear response... 

Applying then the general expression for the first derivative of a perturbation-dependent expectation value, Eq. (3.41), to a component of the molecular magnetic induction B Rk,B) at the position of nucleus K in the presence of an external magnetic induction we obtain expressions for the tensor components of the nuclear magnetic shielding tensor of nucleus K involving first derivatives of the perturbed wavefunction... 

Inserting the perturbation theory expansion of the perturbed wavefunction up to first order, Eq. (3.27), or the expression for the static response function Eq. (3.114) yields the same expressions for the nuclear magnetic shielding tensor and reduced indirect nuclear spin-spin coupling tensor as derived above. 

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