First-quantized formalism

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. 

In the first quantization formalism, one-electron operators are written as... 

In principle, there are two equivalent ways for introducing operators in second quantization. The first is a more physical approach one may (in fact, one must) demand that the expectation value of any observable be the same in the second and the first quantized formalisms. Textbooks on second quantization usually choose this way (see, e.g. Szabo Ostlund 1982, and references therein). Here we shall proceed in a more formal manner, which, however, permits us to construct the second quantized operators, not only to introduce them heuristically. 

The somewhat awkward antisymmetiizing operator necessarily in first quantization is replaced by formal rules for manipulating creation and annihilation operators. 

Expressions in general are easier to manipulate with formal rules, the same derivations in first quantization often require many explicit summation indices. 

Most of this chapter utilizes the first-quantized formulation of the ROMs introduced above. However, some concepts related to separabihty and extensiv-ity are more easily discussed in second quantization, and the second-quantized formalism is therefore employed in Section IE. Introducing an orthonormal spin-orbital basis 1 ) = dj 0), the elements of the p-RDM are expressed directly in second quantization as... 

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 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... 

The eigenfunctions of the zeroth order Hamiltonian define the projection of the DCB equation onto the subspace of electronic solutions. This is a first and necessary step to apply QED theory in quantum chemistry. The resulting second quantized formalism is compatible with the non-relativistic spin-orbital formalism if the connection (unbarred spinors <-> alpha-spinorbitals) and (barred spinors beta spinorbitals) is made. This correspondence allows transfer to the relativistic domain of non-relativistic algorithms after the differences between the two formalism are accounted for. 

In the 1960s and early 1970s, even the formalism of second quantization was shunned by most quantum chemists, even deemed as an unnecessary extravagance. For this reason, we wrote a paper [85] in which we derived the CCD equations using the standard first quantization wave-function formahsm, without any diagrams (for a similar CCSD version, see Ref. [86], written in connection with the appearance of the quadratic Cl [87]). Yet, even here, we tried for too much brevity and compact mathematical notation. For example, the key expression for the disconnected quadruples in terms of doubles, later on usually written in its full form listing all 18 terms (see, e.g. the last Eq. (33) of Ref. [88] note already a much more compact form used in Eq. (11) of Ref. [89]),... 

The selection rules on Av can be extracted by applying the second quantization formalism to Eq. 6.98. In particular, the first-order terms proportional to Qi permit transitions with Avi = 1, the second-order terms in QiQj are responsible for the overtone and combination bands with A Vi + Vj) = 0 or +2, and so on. The symmetry selection rule must simultaneously be satisfied. It is well known to students in organic chemistry that overtone and combination bands are frequently prominent in infrared spectra, and so the second- and higher order terms in Eq. 6.98 are not negligible. 

Among possible approaches, the so-called second quantization plays an important role. The ultimate goal of the second quantized approach to the many-electron problem is to offer a formalism which is substantially simpler than the traditional one in many cases. As a matter of fact, most difficulties of the traditional or first quantized approach arises from the Pauli principle which requires the wave function W of Eq. (1.1) to be antisymmetric in the electronic variables. This is an additional requirement which does not result from the Schrodinger equation and requires a special formalism the using of Slater determinants for constructing appropriate solutions to Eq. (1.1). The Slater determinant is not a very pictorial mathematical entity, and the evaluation of matrix elements over determinantal wave functions makes the first quantized quantum chemistry somewhat complicated for beginners. In the second quantized... 

Another useful feature of the second quantized formalism is that the second quantized representant of the Hamiltonian (and any other physical observable) is independent of the number of electrons, N, in contrast to the first quantized form of the Hamiltonian, Eq. (1.2). Thus, chemical systems containing different numbers of electrons, for example, can be described by one and the same (or very similar) Hamiltonian. 

Being familiar with the formalism of permutations or the properties of the antisym-metrizer operator, one may simplify the first quantized derivation, too. Nevertheless, the second quantization-based consideration remains the simplest... 

By means of the first- and second-order density matrices introduced above it is a trivial task to derive the expression of the electronic energy in the Hartree-Fock theory. The goal is simply to evaluate the expectation value of the Hamiltonian H, which in the second quantized formalism is given by Eq. (4.40) ... 

It is important to observe that the spin labels are not eliminated from the second quantized form of the Hamiltonian. They do not appear in the list of the integrals, however, which corresponds to the fact that the first quantized Hamiltonian is spin-independent and permits one to use the spin-free formalism. But it is essential to realize that creation and annihilation operators cannot be specified merely for spatial orbitals. 

We shall mention that using the mixed second quantized formalism of Ref. [1], already mentioned, it is possible to present the chemical Hamiltonian (32)-(34) in a form in which each term of the Hamiltonian contains only creation and annihilation operators assigned to the corresponding atom or pair of atoms. To save place, we shall illustrate that only by considering the first term of Eq. (33)— all the other terms can be treated analogously. The first term in question is... 

As said above, pd are the marginal distributions of the whole distribution D. To obtain them, the contraction mapping (CM) operation may be performed on D in order to reduce the number of variables from a fixed M number of particles to p, i.e., the order of contraction [26, 41]. In order to define this operation for the GC distribution which has no fixed number of particles, let us first sketch it for the MC and C distributions. For this goal, we introduce the p-RDMs in terms of the p-order replacement operators pe [42] in the second quantization formalism [43]... 

Jeziorski et al, have formulated a first-quantization form of the CCD equations where the pair functions are not expressed in terms of double replacements but as expansions in Gaussian geminals, In the original derivation of the theory, they have employed a spin-adapted formulation in terms of singlet and triplet pairs, but a spin-orbital formalism will be used in the following for the sake of a compact presentation,... 

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. 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

While early work [16, 19] on the CSE assumed that Nakatsuji s theorem [37], proved in 1976 for the integrodifferential form of the CSE, remains valid for the second-quantized CSE, the author presented the first formal proof in 1998 [20]. Nakatsuji s theorem is the following if we assume that the density matrices are pure A-representable, then the CSE may be satisfied by and if and only if the preimage density matrix D satisfies the Schrodinger equation (SE). The above derivation clearly proves that the SE imphes the CSE. We only need to prove that the CSE implies the SE. The SE equation can be satisfied if and only if... 

In order to describe microscopic systems, then, a different mechanics was required. One promising candidate was wave mechanics, since standing waves are also a quantized phenomenon. Interestingly, as first proposed by de Broglie, matter can indeed be shown to have wavelike properties. However, it also has particle-Uke properties, and to properly account for this dichotomy a new mechanics, quanmm mechanics, was developed. This chapter provides an overview of the fundamental features of quantum mechanics, and describes in a formal way the fundamental equations that are used in the construction of computational models. In some sense, this chapter is historical. However, in order to appreciate the differences between modem computational models, and the range over which they may be expected to be applicable, it is important to understand the foundation on which all of them are built. Following this exposition. Chapter 5 overviews the approximations inherent... 

Eq. (351) can be transformed to Eq. (359). Further identifying Ns with 2ti p( ), Eq. (346) becomes identical with Eq. (361). Hence, under certain circumstances the quantized ARRKM theory is equivalent to the rigorous quantum reaction rate theory. A number of remarks are in order. First, assumption (a) is automatically satisfied by definition. Second, assumption (b) implies that Fw in the quantized ARRKM theory be the direct analog of the quantum flux operator in the flux-flux autocorrelation formalism. Third, assumption (c) requires that the action of the operator 0jy(V5 v) at any particular time, say at time zero, is equivalent to the action of the projector P i) at time infinity. Regarding 0vi (V5 v) as the analog... 

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