Occupation-number vector

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. 

Let 0i(r,a), i = 1, m denote a basis of m orthonormal spin orbitals where r are the spatial coordinates and o is the spin coordinate. A Slater determinant is an antisymmetric linear combination of one or more of these spin orbitals. The occupation of a given Slater determinant can be written as an occupation number vector, ln>, where nj is one if spin orbital j is occupied in the Slater determinant and nj is zero if spin orbital (j> is unoccupied. 

The occupation number vectors are basis vectors in an m-dimensional abstract linear vector space, the Fock space, F(m). For a given spin orbital basis, there is a one-to-one mapping between a Slater determinant and an occupation number vector in the Fock space. The occupation number vectors are not Slater determinants they do not have any spatial structure, they are just basis vectors in a linear vector space. Much of the terminology which is used for Slater determinants is, however, used for occupation number m... 

The Fock space F(m) is a sum of subspaces F(m,N) where each subspace F(m,N) contains all occupation number vectors that can be obtained by distributing N electrons into m spin orbitals. The subspace consisting of occupation number vectors with zero electrons contains a single vector, the true vacuum state,... 

For an orthonormal basis of spin orbitals, we define the inner product between two occupation number vectors I n> and I k> as... 

This definition is consistent with the definition of overlap between two Slater-determinants having the same number of electrons. The overlap between Slater determinants having a different number of electrons is not defined. The extension to have a well-defined, but zero, overlap between two occupation number vectors with different numbers of electrons is a special feature of the Fock-space formulation that allows a unified description of systems with a different number of electrons. As a special case of Eq. (1.3), the vacuum state is defined to be normalized... 

The phase factor r(n) is introduced in order to endow the antisymmetry of many-electron wave functions in the Fock space, as we soon will see. The definition that ai operating on an occupation number vector gives zero if spin... 

The action of two operators aj and aj on an occupation number vector with spin orbital i and j, i[Pg.40]

All occupation number vectors in F(m,N) can be obtained from an occupation number vector I n> with N electrons by applying one or several elementary excitation operators on I n>. If a single excitation operator is applied we obtain a single excitation, if two excitation operators are involved, we obtain a double excitation, etc. 

The occupation number vectors are thus the common eigenvectors for the hermitian and commuting set of operators (aj a agag,. -.a am) and there is a one to one correspondence between an occupation vector and a set of eigenvalues for (aj av a ag,- -.a am). This is consistent with the definition of the occupation number vectors as being an orthonormal basis for the Fock space. 

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 annihilation operators are written to the right of the creation operators to ensure that g operating on an occupation number vector with less than two electrons vanishes. Using that the annihilation operators anticommute and that the creation operators anticommute it is easy to show that the parameters g can be chosen in a symmetric fashion... 

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 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 preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

Eq. (4.22) thus gives an operator representation of the mapping from at to at One of the advantages of using the transformation in Eq. (4.22) is that it allows operator manipulations of formulae. Consider for example an occupation number vector I n k> with occupation vector n referring to orbitals in the transformed basis. This vector can be written... 

In the previous sections, the occupation number vectors were specified in terms of the occupation of a set of spin orbitals, and the operators were defined by integrals over spin orbitals multiplied with spin orbital excitation operators. The spin orbitals depend on a continuous spatial coordinate, r, and a discrete spin coordinate ms. The spin coordinate takes two values, so the complete spin basis is spanned by two functions a(ms), a = a, p defined as... 

The occupation number vectors can be written as a product of alpha creation operators, an alpha string, times a product of beta creation operators, a beta string... 

The evaluation of a spin operator times an occupation number vector is faciliated by noting that the core is a singlet spin tensor, since ata ajjj is a singlet spin operator (Eq. 5.25). The action of Sz on I na np> becomes... 

The number of occupation number vectors generated by the action of... 

The action of S on I na np> gives thus a term proportional to I na np> plus a sum over occupation number vectors, where the spin functions of an alpha orbital and a beta orbital have been flipped. Only permutations of the singly occupied orbitals are included. Inanp> is thus in general not an... 

A Fock space corresponding to this set can again be defined as an abstract vector space with basis vectors defined as occupation number vectors I n>, n ... 

The inner product between two occupation number vectors is defined so that for two occupation number vectors with the same number of electrons, the overlap equals the overlap between the corresponding two Slater determinants. Two occupation number vectors with different numbers of electrons are defined to have overlap 0. For two occupation number vectors ln> and lm> with ne and me electrons, respectively, we... 

An annihilation operator times the vacuum state still vanishes so the effect of an annihilation operator times an occupation number vector becomes... 

The above development shows that the effect of operators times an occupation number vector can be evaluated for the case, where the basis is nonorthogonal. The remaining problem in using nonorthogonal orbitals is how to efficiently calculate inner products between occupation number vectors. We will not describe that in any more detail in this book. 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

The occupation number vector can be written as a sequence of O s and I s according to the occupation of a particular one-electron state... 

From the above mentioned relations it is easy to see that the vacuum expectation value of the electronic Hamiltonian (3.4) is zero. The particle-hole formalism implies a redefinition of the vacuum state. Since correlation energy is defined with respect to the Hartree-Fock energy, we redefine the vacuum state as being the occupation-number vector corresponding to the converged HF determinant, the Fermi vacuum. This leads to a redefinition of creation... 

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