Propagators and Second Quantization

The special propagator methods were introduced into quantum chemistry around 1965 by Linderberg and Ohm,17 and during the last two decades there has been a tremendous development in this field.18 If A and B are two arbitrary operators expressed in terms of second quantization, a propagator ( A B))z in their approach is a function of the complex variable z which satisfies the expansion formula ... 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

Here e1 is the fundamental transverse microscopic electric held operator and b is the corresponding magnetic held operator. The superscript on the electric held operator designate its transverse character with respect to the direction of propagation, redundant in the case of the magnetic held as it is intrinsically transverse, namely, divergence-free, since it arises from the curl of a vector potential held a(r). Since the electric held also derives from a(r), we concentrate first on the second-quantized form of this vector potential, which is cast in terms of a summation over radiation modes as follows ... 

Second quantization was frowned upon by many 50 years ago, see e.g. Slater s review [1] of Judd s book. This stylish technique, Slater ibid., has gained acceptance and the formulations from Propagators in Quantum Chemistry [2] have evolved to the forms presented here. Ballhausen and Gray [3] offered their interatomic coupling form as an alternative to the choice by Wolfsberg and Helmholz [4]. It was Fondon who introduced the field dependent couplings in the... 

This edition, a completely rewritten and expanded version of the original, includes second quantization and diagrammatic perturbation theory, symmetric and unitary group methods, new forms of valence bond theory, dynamic properties and response, propagator and equation-of-motion techniques and the theory of intermolecular forces. 

The second factor in (7.45) is the Clebsch-Gordan coefficient, which depends on the rotational quantum numbers J, of the lower and upper levels, and on the orientation of the molecule with respect to the quantization axis (in case of a pump wave this is the propagation direction k, in case of a rr-pump wave it is the direction of the electric vector E), see Fig. 7.24. 

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