The zeroth order Hamiltonian

The zeroth order Hamiltonian, be it relativistic or not, for an iV-electron system in the presence of n fixed nuclei has the form 

In the 4-component relativistic case, the one-electron operators correspond to the Dirac operator in the field of static nuclei 

Most 4-component relativistic molecular calculations are based on the Dirac-Coulomb Hamiltonian corresponding to the choice g = Coulomb The Gaunt term of (173) has been written in a somewhat unusual manner. The speed of light has been inserted in the numerator which clearly displays that the Gaunt term has the form of a current-current interaction, contrary to the 

The reader may note that the general theory of molecular properties of section 2 has been developed with very few references to relativity. This is an important point and signals that the methods used in the relativistic domain has essentially the same structure as in the non-relativistic domain. This point can be emphasized further by rewriting the zeroth order Hamiltonian (169) in second quantization [80] 

Each individual integral is a scalar, no matter whether it is generated from 1-, 

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The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, E, P, Uj, It, Uc, and Ic-The terms of H involving the parameters aj, ac, and bo aie diagonal with respect to both the Ij and Ic quantum numbers, while the f>2 term connects with one another the basis functions with I j = Ij 2, 4- 2. The c terms... 

The eigenfunctions of the zeroth-order Hamiltonian are written with energies. ground-state wavefunction is thus with energy Eg° To devise a scheme by Lch it is possible to gradually improve the eigenfunctions and eigenvalues of we write the true Hamiltonian as follows ... 

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

Since the interaction of the electron with the medium polarization is strong, in the reference model it was usually included in the zeroth-order Hamiltonians determining the Born-Oppenheimer electron states ... 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

Definition of the Zeroth Order Hamiltonian in Multiconfigurational Perturbation Theory (CASPT2). 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

The zeroth order Hamiltonian and the perturbation are, therefore, the following ... 

The firstfew coefficients are shown in Table I. Finally, it should be clear that none of those potential pieces in Eq. (12) are included in the zeroth-order Hamiltonian ftp in Eq. (76) and the entire KS effective potential is treated as the perturbation. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

Consider the excitation of a molecule with the energy-level scheme described. Let ifiE be an eigenfunction of the complete Hamiltonian, H, and zeroth order Hamiltonian, H0, corresponding to the single discrete state, the set of vibronic states of another electronic state, and the fragmentation continuum, respectively. We write,... 

Electronic Predissociation. We have developed an approach based on the BO approximation (29). In this development the eigenfunctions of the zeroth-order Hamiltonian are BO wave-functions. As was noted above for direct photodissociation, a... 

Hamiltonian proposed by Muller and Plesset gives rise to a very successful and efficient method to treat electron correlation effects in systems that can be described by a single reference wave function. However, for a multireference wave function the Moller-Plesset division can no longer be made and an alternative choice of B(0> is needed. One such scheme is the Complete Active Space See-ond-Order Perturbation Theory (CASPT2) developed by Anderson and Roos [3, 4], We will briefly resume the most important definitions of the theory one is referred to the original articles for a more extensive description of the method. The reference wave function is a CASSCF wave function that accounts for the largest part of the non-dynamical electron correlation. The zeroth-order Hamiltonian is defined as follows and reduces to the Moller-Plesset operator in the limit of zero active orbitals ... 

Resonances in the absorption spectrum are the fingerprints of the eigenenergies of the binding part of the zeroth-order Hamiltonian (i.e., the Hamiltonian in the absence of coupling between the two diabatic states). 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

It is always possible to divide the total Hamiltonian Xinto a major part 3C (the zeroth-order Hamiltonian) and a perturbation XX ... 

In this case, the zeroth-order Hamiltonian is chosen to represent the vibrational energy of the anharmonic oscillator ... 

Because we have chosen the potential function Vrj(R) to be independent of the reduced mass in the zeroth-order Hamiltonian, the isotopic dependences of the various terms in (7.179) are quite explicit. The leading term Dve is proportional to /i 2 and the vibrational dependent term aDv is proportional to /jr5/2. 

This is an important result. The first term leads to the rotational eigenvalues, whilst the second term describes the rotational electronic coupling and, as we shall see, contributes to the rotational magnetic moment and the spin rotation interaction. The third term is small and can be neglected for states where A = 0. We have omitted the electron kinetic energy term from (8.101) because it is part of the zeroth-order Hamiltonian which determines the electronic eigenvalues and eigenfunctions. 

The effects of the off-diagonal terms when folded-in by perturbation theory are of two types. They can either produce operators of the same form as those which already exist in the Hamiltonian constructed from the Azl = 0 matrix elements (the zeroth-order Hamiltonian), or they can have a completely novel form. A good example of the former type is the second-order contribution to the rotational constant which arises from admixture of excited and A states,... 

The first-order part describes the nuclear and the second-order the electronic contribution to the molecular moment of inertia. The A-doubling terms, on the other hand, have no counterpart in the zeroth-order Hamiltonian. For a 2 n state, the operator form is... 

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