Operator unperturbed

Each electron in the system is assigned to either molecule A or B, and Hamiltonian operators and for each molecule defined in tenns of its assigned electrons. The unperturbed Hamiltonian for the system is then 0 = - A perturbation XH consists of tlie Coulomb interactions between the nuclei and... 

This method [ ] uses the single-configuration SCF process to detennine a set of orbitals ( ).]. Then, using an unperturbed Flamiltonian equal to the sum of the electrons Fock operators // = 2 perturbation... 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

The MoIIer-PIesset perturbation method (MPPT) uses the single-eonfiguration SCF proeess (usually the UHF implementation) to first determine a set of LCAO-MO eoeffieients and, henee, a set of orbitals that obey F( )i = 8i (jii. Then, using an unperturbed Hamiltonian equal to the sum of these Foek operators for eaeh of the N eleetrons =... 

This Foek operator is used to define the unperturbed Hamiltonian of Moller-Plesset perturbation theory (MPPT) ... 

The first-order energy involves only the perturbation operator and the unperturbed wavefunction. In an HF-LCAO treatment, the integrals would be over the LCAOs, and this implies a four-index transformation to integrals over the basis functions. 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

The formulas for higher-order conections become increasingly complex. The main point, however, is that all corrections can be expressed in terms of matrix elements of the perturbation operator over the unperturbed wave functions, and the unperturbed energies. 

So far the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamilton operator must be selected. The most common choice is to take this as a sum over Fock operators, leading to Mdller-Plesset (MP) perturbation theory. The sum of Fock operators counts the (average) electron-electron repulsion twice (eq. (3.43)), and the perturbation becomes... 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

Where // is the complete Hamiltonian operator for the unperturbed system and the usual quantum-mechanical integrals over all space are indicated. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

In order to define the notation which we will use from now on, let us consider the application of the perturbation theory to a system which has a perturbed hamiltonian H composed by an unperturbed one, H", plus a perturbation operator A.V, where A, () ... 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

In many applications there is no second-order term in the perturbed Hamiltonian operator so that zero. In such cases each unperturbed... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

In reality, this term is not small in comparison with the other terms so we should not expect the perturbation technique to give accurate results. With this choice for the perturbation, the Schrodinger equation for the unperturbed Hamiltonian operator may be solved exactly. 

The unperturbed Hamiltonian operator is the sum of two hydrogen-like Hamiltonian operators, one for each electron... 

In fin problem of interest here, the Hamiltonian in Bq. (62) can be decomposed into a time-independent, unperturbed part and a much smaller, time-dependent operator H (t). Then, die Hamiltonian becomes to first oiler... 

In this interpretation Q is the number of linearly independent eigenfunctions of the unperturbed Hamiltonian in the interval AE. From (36) the microcanonical average of an observable represented by the operator A in an arbitrary basis, is... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

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