Zeroth-order operators

The zeroth-order operator will be defined by the following eigenvalue equation ... 

In practical applications of the sapt approach to interactions of many-elect ron systems, one has to use the many-body version of sapt, which includes order-by-order the intramonomer correlation effects. The many-body SAPT is based on the partitioning of the total Hamiltonian as H = F+V+W, where the zeroth-order operator F = Fa + Fb is the sum of the Fock operators for the monomers A and B. The intermolecular interaction operator V = H — Ha — Hb is the difference between the Hamiltonians of interacting and noninteracting systems, and the intramonomer correlation operator W = Wa + Wb is the sum of the Moller-Plesset fluctuation potentials of the monomers Wx — Hx — Fx, X — A or B. The interaction operator V is taken in the non-expanded form, i.e., it is not approximated by the multipole expansion. The interaction energy components of Eq. (1) are now given in the form of a double perturbation series,... 

In Eq. (3) we find orbital energy differences in the denominator. This is due to the fact that we took the Fock operator as the zeroth-order operator, as did Mpller and Plesset [62] in 1934. An alternative zeroth-order operator is due to Epstein [70] saved from oblivion by Nesbet [71]. This operator can be written as [72]... 

The unperturbed operator — in this context then also called zeroth-order operator — is the one-electron Dirac Hamiltonian of Eq. (8.64)... 

In this representation both the Hamiltonian and the metric partition transparently into zeroth-order operators and a perturbation. 

Employing simplifications arising from the use of asymptotic forms of the electronic basis functions and the zeroth-order kinetic energy operator, we obtain... 

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 observation of nitration at a rate independent of the concentration and nature of the aromatic excludes AcONOa as the reactive species. The fact that zeroth-order rates in these solutions are so much faster than in solutions of nitric acid in inert organic solvents, and the fact that HNO3 and H2NO3+ are ineffective in nitration even when they are present in fairly lai e concentrations, excludes the operation of either of these species in solutions of acetyl nitrate in acetic anhydride. 

The C matrix, the columns ofwhich, Cj(, are the eigenvectors of H, is normally not too different from the matrix defined above. However, the QDPT treatment, applied either to an adiabatic or to a diabatic zeroth-order basis, is necessary in order to prevent serious artefacts, especially in the case of avoided crossings [27]. The preliminary diabatisation makes it easier to interpolate the matrix elements of the hamiltonian and of other operators as functions of the nuclear coordinates and to calculate the nonadiabatic coupling matrix elements ... 

If the coefficients Ckk(Q) and c xiQ) and the operators A x are sufficiently small, the summation on the left-hand side of equation (10.18) and Ckk(Q) in (10.19) may be neglected, giving a zeroth-order equation for the nuclear motion... 

The perturbation operators must be modified in accordance with the new definition of the zeroth-order electron states ... 

Together with Eq. (66), this equation describes exactly the linear response of the system to an external field, with arbitrary initial conditions. Its physical meaning is very simple and may be explained precisely as for Eq. (66) 32 the evolution of the velocity distribution results in two effects (1) the dissipative collisions between the particles which are described by the same non-Markoffian collision operator G0o(T) 35 1 the field-free case and (2) the acceleration of the particles due to the external field. As we are interested in a linear theory, this acceleration only affects the zeroth-order distribution function It is... 

Application of the generalized operator transform yields for a single nucleus the following nuclear transition probability for zeroth-order base functions45 ... 

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)... 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

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 ... 

In Fig. 30.3, the thermodynamic efficiencies of a constant-pressure Bra3don cycle, a constant-volume Humphery cycle (which approximates a PDE cycle), and a true detonation Chapman-Jouguet (CJ) cycle for a typical hydrocarbon fuel are compared [11]. Though the constant-volume cycle shows substantial efficiency advantage, this zeroth order comparison cannot be taken as the correct quantitative comparison, since PDE operates in a pulsed transient mode. How-... 

In order to define Hq such that 4tsoAGP is an eigenfunction, Hq was split into a sum of operators Hk- These operators Hk are defined so that HkSk = Ef gx (Eq. (14)). Then the zeroth-order energy is merely W = H Hq. 

---