Normal form, Hamiltonian operator

As the operator adjy is diagonal, we can proceed by examining if a simple monomial will be included in the normal-form Hamiltonian or not. Specifically, a monomial of degree n + 2,... 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

The theory is based on an optimized reference state that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4> // < >)is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is... 

This result is easily generalized the normal-ordered form of an operator is simply the operator itself minus its reference expectation value. For the Hamiltonian example, above, the normal-ordered Hamiltonian is just the Hamiltonian minus the SCF energy (i.e., may be considered to be a correlation operator). Owing to its considerable convenience for coupled cluster and many-body perturbation theory analyses, this conventional form of f given in Eq. [105] is adopted for the remainder of this chapter. 

The normal product form of the Hamiltonian operator is obtained by the use of the time-independent Wick theorem5-27 (also known as Wick s first theorem29) i.e.,... 

The second form, the normal order of the generator product, shows that the operator also preserves and eigenvalues since it is constructed from operators that do so. The expansion of the Hamiltonian operator in this spinpreserving operator basis shows that the Hamiltonian operator itself must preserve the and eigenvalues of the wavefunctions on which it acts. The definition of the operator results in the identities... 

The optimization of the MCSCF wavefunction requires the construction of matrix elements of the Hamiltonian operator in the CSF expansion basis. The energy expectation value of the Hamiltonian operator of Eq. (115) may be written, assuming real normalized CSF expansion coefficients, in the forms... 

The three-dimensional algebraic model can reproduce, in detail, the aforementioned classification of the rovibrational Hamiltonian operator in partials fi22- To achieve this goal, we start by writing a compact form of this operator in its usual (normal coordinate) notation ... 

The general theorem of the previous section will now be applied to the problem under discussion. If the kinetic energy were expressed in terms of the Eulerian angles 6, , % and the normal coordinates Qk, together with the conjugate momenta p, p, p and Pk, the Hamiltonian operator would assume the form of Eq. (6),... 

Analogous relations hold for three-fermion terms, which also occur in the transformed Hamiltonian. After complex application of the Wick s theorem on all fermion operators we get the normal form of the Hamiltonian in the general representation. 

This leads immediately to the Hamiltonian operator in a form which is of primary importance for perturbation treatments and many-body formalism. This form of the Hamiltonian operator is termed the normal product form. It is usually written as follows ... 

The most important property of the Hamiltonian operator, for our purposes, is that its eigenfunctions Wi,. .., may be assumed to form a complete orthogonal set (Kato, 1951) if they also belong to class 1 we may normalize in the usual way and write... 

The operator in parentheses is known as the Hamiltonian operator and normally is designated by the symbol H. Hence, we can write the Schrodinger equation in eigenvalue form as... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

As concerns cluster amplitudes, if we employ the exact Hamiltonian in the normal-ordered-product form (31) with the /i-th configuration as a Fermi vacuum, the basic equation for the single-root wave operator (25) takes the form... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

Note that all the above expressions characterize the effective Hamiltonian formalism per se, and are independent of a particular form of the wave operator U. Indeed, this formalism can be exploited directly, without any cluster Ansatz for the wave operator U (see Ref. [75]). We also see that by relying on the intermediate normalization, we can easily carry out the SU-Ansatz-based cluster analysis We only have to transform the relevant set of states into the form given by Eq. (16) and employ the SU CC Ansatz,... 

The problem with these equations is that they correspond to infinite different Hamiltonians so that the solutions for different electronic quantum numbers are incommensurate. To do away with these objections, use instead the complete set of functions rendering the kinetic energy operator Kn diagonal. The set, within normalization factors, is fk(Q) exp(ik Q) k is a vector in nuclear reciprocal space. Including the system in a box of volume V, the reciprocal vectors are discrete, ki, and the functions f (Q) = (1/Vv) exp(iki Q) form an orthonormal set with the completeness relation 8(Q-Q ) = Si fi(Q) fi(Q )- The direct product set ( )j(q)fki(Q) is complete. The matrix elements of eq. (8) reads ... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

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