Quantization, first

The notation used in this book is in terms of first quantization. The electronic Hamilton operator, for example, is written as (eq. (3.23))... 

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). 

There are several advantage of second quantization over first quantization. 

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. 

Among the several 2-RDM-oriented methods that have been developed for the study of chemical systems, one of the most recent and promising techniques is based on the iterative solution of the second-order contracted Schrodinger equation (2-CSE) [1, 6, 15, 18, 36, 45-60, 62-65, 68, 70, 79-85, 103-111]. The 2-CSE was initially derived in 1976 in first quantization in the works of Cho [103], Cohen and Erishberg [104, 105], and Nakatsuji [106] and later on deduced in second quantization by Valdemoro [45] through the contraction of... 

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 importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

The orbital representation is not used in most of the recent work on computational methods based on diagonal elements of density matrices. This is partly for historical reasons—most of the work has been done by people trained in density functional theory—and partly this is because most of the available kinetic energy functionals are known only in first-quantized form. For example, the popular generalized Weisacker functional [2, 7-11],... 

In 1907. Einstein showed that at extremely low temperatures, the atoms of a solid don t have sufficient energy to jump to the first quantized energy level, which is a relatively large jump. The solid, therefore, may be exposed to small increments of heat without any increase in thermal motion. This lowers the ability of the solid to absorb heat, which means that its entropy is also lower. For practically all materials, this quantum effect only occurs at extremely low temperature. A dramatic exception is diamond, which because of this quantum effect resists the absorption of energy even at room temperature (Table 9.2). Diamond is special for many reasons, including its status as a room-temperature "quantum solid"... 

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 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... 

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

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... 

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 quantum mechanical operators can be divided according to spatial and spin properties. In first quantization a pure spatial (spin free) operator Fc does not change the spin functions so Fc commutes with the spin function... 

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... 

Our next step is to minimize the energy of the total system with respect to the density and thereby we are able to define an effective Kohn-Sham (KS) operator. In first quantization, the KS operator is divided into a vacuum and a coupling contribution... 

In Eq. (18), we recognize the first quantization fcth-order electronic multipole moment operator (r - Ro). An analogous expansion of r - Rm 1 would yield the nuclear multipole moment operator (Rm — Rq). The higher-order multipole moments generally depend on the choice of origin however, to simplify the notation, we omit any explicit reference to this dependence. The partial derivatives in Eq. (18) are elements of the so-called interaction tensors defined as... 

The notation used in this book is in terms of first quantization. The electronic Hamilton... 

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... 

Below the Debye temperature, only the acoustic modes contribute to heat capacity. It turns out that within a plane there is a quadratic correlation to the temperature, whereas linear behavior is observed for a perpendicular orientation. These assumptions hold for graphite, which indeed exhibits two acoustic modes within its layers and one at right angles to them. In carbon nanotubes, on the other hand, there are four acoustic modes, and they consequently differ from graphite in their thermal properties. StiU at room temperature enough phonon levels are occupied for the specific heat capacity to resemble that of graphite. Only at very low temperatures the quantized phonon structure makes itself felt and a linear correlation of the specific heat capacity to the temperature is observed. This is true up to about 8 K, but above this value, the heat capacity exhibits a faster-than-Unear increase as the first quantized subbands make their contribution in addition to the acoustic modes. 

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