First-quantized operators

The aja, operator tests whether orbital i exists in the wave function, if that is the case, a one-electron orbital matrix element is generated, and similarly for the two-electron terms. Using the Hamiltonian in eq. (C.6) with the wave function in eq. (C.4) generates the first quantized operator in eq. (C.3). 

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... 

The factors g can be identified by calculating the matrix elements of g, cnlgl m>, between two occupation number vectors and requiring that the obtained expressions should be equal to the matrix elements between the corresponding Slater-determinants of the first quantization operators. We consider four different cases for these matrix elements... 

The product of the two first quantization operators Oj 02 can be separated into a one-electron part and a two-electron part... 

The above development indicates that commutation relationships that hold for first quantization operators do not necessarily hold for second quantization operators in a finite one-electron basis. Consider the canonical commutators... 

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... 

The RS formulas for the energy expansion are well known and are given in many places (e.g., Ref. 22). A thorough development of the wave-reaction operator perturbation theory has been presented by Low-din.23 Using conventional first quantized operators, we may write down the expressions for the nth-order energy E(n), for instance, as... 

Having now seen how state vectors that are in one-to-one correspondence with /V-eIcctron Slater determinants can be represented in terms of Fermion creation and annihilation operators, it still remains for us to show how to express one- and two-electron operators in this language. The second-quantized version of any operator is obtained by simply demanding that the operator, when sandwiched between ket vectors of the form [ r vac>, yield exactly the same result as arises in using the first quantized operator between corresponding Slater determinant wavefunctions. For an arbitrary one-electron operator, which in first-quantized language is 5] = i /( i) second quantized equivalent is... 

The operator presented above is essentially the first-quantized operator. The second-quantized form is... 

In Box 1.1, we summarize the fundamentals of the second-quantization formalism. In Section 1.4, we proceed to discuss the second-quantization representation of standard first-quantization operators such as the electronic Hamiltonian. 

Expectation values correspond to observables and should therefore be independent of the representation given to the operators and the states. Since expectation values may be expressed as sums of matrix elements of operators, we require the matrix element of a second-quantization operator between two ON vectors to be equal to its counterpart in first quantization. An operator in the Fock space can thus be constructed by requiring its matrix elements between ON vectors to be equal to the corresponding matrix elements between Slater determinants of the first-quantization operator. 

First-quantization operators conserve the numbCT of electrons. Following the discussion in Section 1.3, such operators are in the Fock space represented by linear combinations of operators that contain an equal number of creation and annihilation rqierators. The explicit form of these number-conserving operators depends on whether the first-quantized operator is a one-electron operator or a two-electron operator. One-electron operators are discussed in Section 1.4.1 and two-electron operators in Section 1.4.2. Finally, in Section 1.4.3 we consider the second-quantization representation of the electronic Hamiltonian operator. 

From the construction of the second-quantization operators, it is clear that the first-quantization operator -I- bBf, where a and b are numbers, is represented by oA bB. The standard relations... 

In these expressions, square brackets around a first-quantization operator represent the one-electron integral of this operator in the given basis. This somewhat cumbersome notation is adopted for this discussion to make the dependence of the integrals on the first-quantization operators explicit. In Section 1.8, the commutator between the two excitation operators is shown to be... 

In Box 1.2, we summarize some of the characteristics of operators in the first and second quantizations. The dependence on the spin-oibital basis is different in the two representations. In first quantization, the Slater determinants depend on the spin-orbital basis whereas the operators are independent of the spin orbitals. In the second-quantization formalism, the ON vectors are basis vectors in a linear vector space and contain no reference to the spin-orbital basis. Instead, the reference to the spin-orbital basis is made in the operators. We also note that, whereas the first-quantization operators depend explicitly on the number of electrons, no such dependence is found in the second-quantization operators. 

The integrals entering the second-quantization operator / vanish for c posite spins since the first-quantization operator is spin-free ... 

A number of first-quantization operators such as the fine-structure and hyperfine-stracture operators affect both the spatial and spin parts of the wave function. As an example, we here consider the effective spin-orbit interaction operator... 

In our discussion, we have so far examined the electron density in the sjrin-orbital and orbital spaces. Let us now consider the electron density in ordinary space. Of particular interest are the expectation values of operators that probe the presence of electrons at particular points in space. Thus, the one-electron first-quantization operator in the form of a linear combination of Dirac delta functions... 

Chapters 1-3 introduce second quantization, emphasizing those aspects of the theory that are useful for molecular electronic-structure theory. In Chapter 1, second quantization is introduced in the spin-orbital basis, and we show how first-quantization operators and states are represented in the language of second quantization. Next, in Chapter 2, we make spin adaptations of such operators and states, introducing spin tensor operators and configuration state functions. Finally, in Chapter 3, we discuss unitary transformations and, in particular, their nonredundant formulation in terms of exponentials of matrices and operators. Of particular importance is the exponential parametrization of unitary orbital transformations, used in the subsequent chapters of the book. 

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