Operators second-quantized representation

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

The recipe for constructing a second quantization representation of a one-electron operator is thus to use Eq. (2.2) with the integrals of Eq. (2.4). 

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 second quantization representation of the electronic hamilton operator is thus... 

The second quantization representation of a spin-free one-electron operator is... 

Let us now consider a first quantization operator, hc, that only works in the spin space, so Eq. (5.20 holds. The second quantization representation, h, can be written... 

Operators and wave functions in second-quantization representation... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

Generally speaking, the representation in terms of occupation numbers is considered to be an independent quantum-mechanical representation, distinct from the coordinate (or momentum) one. In that case, the occupation numbers for one-particle states are dynamic variables, and operators are the quantities that act on functions of these variables. In this section, second-quantization representation is directly related to coordinate representation in order that in what follows we may have a one-to-one correspondence between quantities derived in each of these representations. 

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. 

Utilization of the tensorial properties of the electron creation and annihilation operators allows us to obtain expansions in terms of irreducible tensors of any operators in the second-quantization representation. So, using the Wigner-Eckart theorem (5.15) in (14.11) and (14.12), then coupling ranks of second-quantization operators by (5.12) and utilizing (14.10), we can represent one-shell operators of angular momentum in the irreducible tensor form... 

For another unit tensorial operator Vkl (5.28) in the second-quantization representation we have... 

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... 

This operator, too, is a scalar in the space of total angular momentum for an electron. Tensors in this space are, for example, the operators of electric and magnetic multipole transitions (4.12), (4.13), (4.16). So, the operator of electric multipole transition (4.12) in the second-quantization representation is... 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

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. 

This formula relates the submatrix element of the creation operator to the CFP. The last expression fully corresponds to similar relations in [12, 96]. The only exception is the monograph [14], where formula (16.4), according to relation (2.8) in the same work, differs from (15.21) by the phase factor (— 1). This difference is explained by the fact that in [14] the wave function Ismp) that corresponds to creation operator afsl appears without phase in the last row of the determinant, and not in its first row, as defined earlier by (2.6). As a consequence, although in the second-quantization representation the explicit form of one-determinant functions is not used, one should have in mind the phase convention for... 

In Chapter 14 we derived, in the second-quantization representation, two different forms of the expressions for that operator - (14.61) and (14.63). To begin with, we consider expression (14.63) in which we, by (15.49), go over to triple tensors. Then, after some transformations and coupling the momenta in quasispin space, we arrive at... 

Since the wave functions with N > v can be found from the wave functions with N = v using (16.1), in the second-quantization representation it is necessary to construct in an explicit form only wave functions with the number of electrons minimal for given v, i.e. IolQLSMq = —Q). But even such wave functions cannot be found by generalizing directly relation (15.4) if operator cp is still defined so that it would be an irreducible tensor in quasispin space, then the wave function it produces in the general case will not be characterized by some value of quantum number Q v). This is because the vacuum state 0) in quasispin space of one shell is not a scalar, but a component of a tensor of rank Q = l + 1/2... 

The above relationships describe the behaviour of tensors (23.52) and their tensorial products. A further task is to express the operators corresponding to physical quantities in terms of these tensors. Specifically, two-particle operator (23.27) is expanded by going over to the second-quantization representation and coupling the quasispin ranks... 

A iV-electron vacancy (hole) in a shell may be denoted as nl N = ni4l+2-N (see aiso Chapters 9, 13 and 16). As we have seen in the second-quantization representation, symmetry between electrons and vacancies has deep meaning. Indeed, the electron annihilation operator at the same time is the vacancy creation operator and vice versa instead of particle representation hole (quasiparticle) representation may be used, etc. It is interesting to notice that the shift of energy of an electron A due to creation of a vacancy B l is approximately (usually with the accuracy of a few per cent) equal to the shift of the energy of an electron B due to creation of a vacancy A l, i.e. 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression... 

The results of response theory are most conveniently cast into a formalism based on second quantization. In a second quantized representation of (66), the spin-operators will transfer to triplet excitation operators which are weighted by the integrals over the orbital parts. The -component will have the form... 

A unique feature of the occupation number representation is that the number of electrons does not appear in the definition of the Hamiltonian operator in this form as it does in the wavefunction form. This is because all of the occupation information resides in the bras and kets. This is true for any operator in second quantized form. This feature is used to advantage in theories that allow the number of particles to change, and to a more limited extent in the calculation of electron affinities and ionization potentials. It is less important to the MCSCF method but it is useful to remember that the bras and kets contain all of the occupation information. Other details of the wavefunction, such as the AO and MO basis set information, are included in the integrals that are used as expansion coefficients in the second quantized representation of the operator. 

Energy operator for a molecular crystal with fixed molecules in the second-quantization representation. Paulions and Bosons... 

All operators in the second-quantization representation act on functions depending on occupation numbers. The occupation number operator will be denoted by Nnf. It is diagonal in the occupation number states representation... 

Making use of the relations (3.11) and (3.14) we obtain the crystal energy operator (3.1) in the second-quantization representation... 

The above relations can be obtained by making use of the expressions for the operators n and II in the coordinate representation (see eqns 3.131 and 3.132). The commutation relations cannot depend on the choice of representation and will also be valid for the operators in the second-quantization representation. Thus, substituting (3.135) and (3.139) into the relations (3.140) we obtain the desired sum rule... 

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