Matrix elements second quantization

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

Other two-electron operators are the mass-polarization and the spin-orbit coupling operator. A two-electron operator gives non-vanishing matrix elements between two Slater determinants if the determinants contain at least two electrons and if they differ in the occupation of at most two pairs of electrons. The second quantization representation of a two-electron operator must thus have the structure... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

In the manipulation of operators and matrix elements in second quantization the commutator... 

An approach to constructing CSFs and matrix elements of the Hamiltonian that initially appears quite different from the symmetric group approach can be developed by considering the second-quantized form of the Hamiltonian. If we have an orthonormal... 

This survey of methods for obtaining configuration state functions has necessarily been brief, since the topic could easily occupy a course of its own. However, we have treated the methods in common use, and much additional material on second quantization techniques and matrix element evaluation will be covered elsewhere. 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

Equations (13.22) and (13.23) define the second-quantization form of an operator corresponding to a physical quantity, if its matrix elements are known in coordinate representations ((13.24) and (13.25)). Specifically, the operator of the total number of particles in a system will be... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

In this chapter we have found the relationship between the various operators in the second-quantization representation and irreducible tensors of the orbital and spin spaces of a shell of equivalent electrons. In subsequent chapters we shall be looking at the techniques of finding the matrix elements of these operators. 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

Formula (17.16) is the most general form of the two-electron matrix element in which all four one-electron wave functions have different quantum numbers. We shall put it into general formula (13.23), whereupon the creation and annihilation operators will be rearranged to place side by side those second-quantization operators whose rank projections enter into the same Clebsch-Gordan coefficient. Summing over the projections then gives... 

The method presented enables expressions to be found for submatrix elements of irreducible tensorial products of second-quantization operators for configurations of any complexity. This method provides a unified approach both to diagonal and non-diagonal (relative to the configuration) matrix elements of operators of physical quantities. 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

In order to calculate the spin-angular parts of matrix elements of the two-particle operator (1) with an arbitrary number of open shells, it is necessary to consider all possible distributions of shells upon which the second quantization operators are acting. In [2] they are found to be grouped into 42 different distributions, subdivided into 4 different classes. This also explains why operator (1) is written as the sum of four complex terms. The first term represents the case when all second-quantization operators act upon the same shell (distribution 1 in [2]), the second describes the situation when these operators act upon the two different shells (distributions 2-10), third and fourth are in charge of the interactions upon three and four shells respectively (distributions 11-18 and 19-42). Such expression is particularly convenient to take into account correlation effects, because it describes all possible superpositions of configurations for the case of two-electron operator. 

Second Quantization Photodissociation Hamiltonian, if we consider a system containing many molecules and fragments, it is convenient to use second quantization formalism. We have introduced above the matrix element for photodissociation (see eqs. 50 and 53-57). Based on it, one can write the total... 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

The coefficients Ck are real and Skk1 =< k k > is the overlap of normalized VB diagrams with identical electron distributions rij. Normalization illustrates the general problem of finding matrix elements between correlated states. We express an operator in second-quantized notation and consider exact eigenstates i> > and x > that may be in the same or different symmetry subspaces. The matrix elements Akk of A are obtained as shown in (11) to give... 

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