Wave functions in second-quantization representation

The traditional description of the wave function of a shell of equivalent electrons was presented in Chapter 9. Here we shall utilize the second-quantization method for this purpose. In fact, the one-electron wave function is 

Operator (p in (15.4) can be represented as one irreducible tensorial product of N creation operators only if there are no degenerate terms. So, at N = 3, we have 

For this vacuum average to be computed, all the annihilation operators will have to be placed to the right of the creation operators. The result can be expressed in terms of 3nj-coefficients. 

If an Z3 configuration 253+1L3 term does not recur, then at different allowed values of L2 and S2 at which (15.5) does not vanish, we shall obtain, up to the phase, the same antisymmetric wave function. Otherwise, operator (15.5) may no longer be used, since the different values of L2S2 do not separate the degenerate terms. To see that this is so, it is sufficient to consider the vacuum average 

In the general case, the second-quantized operator p(/N)a LS) must be some linear combination of irreducible tensorial products of electron creation operators. The combination must be selected so that a classification of states according to additional quantum numbers be provided for. Without loss of generality, all the numerical coefficients in the linear combinations can be considered real. Then, from (14.14), we can introduce the operators 

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

Here is the fragment wavefunction and iJ)q is the Q-state wave-function in second quantization representation. Further, as an example, we have limited consideration here to systems of integral total spin Fermi systems can be treated in a similar way. 

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. 

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... 

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... 

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. 

Now consider that, at certain time to, the system of aU electrons is in the antisymmetric state. Group theory states that the system will be permanently in the antisymmetric state. Thus, instead of N classes of solutions, only one has to be considered. Then, one can make a proper formulation that considers only the space of antisymmetric wave functions. As shall be seen in a forthcoming section, this formulation is called second quantization representation and simplifies several solutions of the many-particle Schroedinger equation. Further, when no external magnetic field is present, there is spin symmetry and the eigenstates are classified according to the eigenstates of and S. ... 

Let us turn now to the problem of creating more than two particles consider a many-electron one-determinantal wave function in its second quantized representation. As is easily seen from the previous example for the two-electron case, a many-electron one-determinantal wave function is constructed by successive application of creation operators on the vacuum state. It will be useful though to consider this problem from a somewhat different point of view. 

We are already familiar with the second quantized form of wave functions. In this section the second quantized form of various quantum-mechanical operators is going to be established. Then we shall be in a position to undertake any quantum-mechanical analysis in this representation. 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

In second quantization, the Pauli antisymmetry principle is incorporated through the algebraic properties of the creation and annihilation operators as discussed in Chapter 1. We note that, in density-functional theory (which bypasses the construction of the wave function and concentrates on the electron density), the fulfilment of the A -representability condition on the density represents a less trivial problem. A density is said to be N-representable if it can be derived from an antisymmetric wave function for N particles [1]. 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

The particle number representation is conceptually very important because, strictly speaking, the abstract wave functions given in this representation serve as the carrier space of the second quantized creation operators. In other words, the creation operators act on the particle-number represented wave functions. 

We emphasize again that the symbol o does not mean equality in the mathematical sense because of the different Hilbert spaces considered. The wave function O on the left-hand side of Eq. (2.53) is represented in the L2 function space (or, in the I2 space in the case of a finite basis), while the second quantized wave function on the right-hand side of Eq. (2.53) makes use of the particle number representation. In a given basis, however, there is a one-to-one correspondence between the two representations. This permits one to apply the above correspondences [Eqs. (2.52)-(2.54)] to rewrite any first quantized wave function to the second quantized language or vice versa. 

Equations (4.27) and (4.38) enable us to represent any one- or two-electron operator in quantum chemistry. We know already the representation of the wave functions according to Eqs. (2.52)-(2.54), and also the basic algebraic properties Eqs. (2.48)-(2.50) for the creation and annihilation operators. This is practically all one needs to know in order to undertake any elementary analysis in quantum chemistry using the language of second quantization. 

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