Complete orthonormal set

We now introduce a second complete orthonormal set of functions v, which we shall take to be functions of t, vt(t). First, however, we merely consider the set (0). Since they are by definition constant in time, they behave like the u, and all preceding equations can be written in the v(0)-representation. A transformation of the functions has the form... 

If the hamiltonian is truly stationary, then the wx are the space-parts of the state function but if H is a function of t, the wx are not strictly state functions at all. Still, Eq. (7-65) defines a complete orthonormal set, each wx being time-dependent, and the quasi-eigenvalues Et will also be functions of t. It is clear on physical grounds, however, that to, will be an approximation to the true states if H varies sufficiently slowly. Hence the name, adiabatic representation. 

If we expand the state function ja ) in terms of the complete, orthonormal set... 

To prove the variation theorem, we assume that the eigenfunctions 0 form a complete, orthonormal set and expand the trial function 0 in terms of that set... 

We assume in this section and in Section 10.2 that equation (10.6) has been solved and that the eigenfunctions Q) and eigenvalues k(Q) are known for any arbitrary set of values for the parameters Q. Further, we assume that the eigenfunctions form a complete orthonormal set, so that... 

A complete orthonormal set of n-component vectors is a set vi,v2,---, vn, satisfying the orthonormality relations... 

The eigenvalue-eigenvector decomposition of a Hermitian matrix with the complete orthonormal set of eigenvectors Vi and eigenvalues A, is written as... 

Since H is Hermitian, the eigenvectors Vj of H form a complete orthonormal set and the vector representing a general state at t = 0 may be expressed as a linear superposition of these eigenvectors, (0) = CjVj, ... 

Let us begin by sketching the general polyatomic formulation of hybridization from the NBO viewpoint. A general hybrid h/A) on atom A can be expanded in the complete orthonormal set of NAOs, (A) on this atom ... 

Except for the initial AO —NAO transformation, which starts from non-orthogonal AOs, each step in (3.38) is a unitary transformation from one complete orthonormal set to another. Each localized set gives an exact matrix representation of any property or function that can be described by the original AO basis. 

The solution of the unperturbed Hamiltonian operator forms a complete orthonormal set. The perturbed Schrodinger equation is given by... 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

Atomic units are used. Here and in the following x = (r, s) stands for the combined spatial and spin coordinates, r and s, respectively. The SOs 0,(x) constitute a complete orthonormal set of single-particle functions. 

Let i//, be an asymmetric-top wave function. A convenient complete orthonormal set to use here is the symmetric-top wave functions, which are functions of the same coordinates (the Eulerian angles) and satisfy the same boundary conditions as the asymmetric-top functions ... 

The wave functions have the form (5.54), but since Pc does not commute with H, we cannot separate out a chi factor the Schrodinger equation is not separable, and we will try another method of dealing with the problem. We saw in Section 2.3 that the eigenvalues of an operator H can be found by expanding the unknown eigenfunctions in terms of some known complete orthonormal set [Pg.361]

As before, we assume that the BO states form a complete orthonormal set such that... 

We shall also assume that the true eigenstates of the total molecular Hamiltonian Hel form a complete orthonormal set,... 

The changes in structure that must occur create a barrier to electron transfer. In order to understand the origin of the barrier and to treat it quantitatively, it is necessary to recall that the structural changes at each reactant can be resolved into a linear combination of its normal vibrational modes. The normal modes constitute a complete, orthonormal set of molecular motions into which any change in intramolecular structure can be resolved. 

Returning to equation (25), evaluation of the total vibrational overlap integral, (Xj X7), is less formidable than it appears. The vibrational wavefunctions are a complete orthonormal set for which ( 1 0 )= where S is the Kronecker delta. For the vast majority of normal modes, S (and AQe) = 0. For these modes the vibrational overlap integrals become (yjy,/) = 1 if v = v, and = 0 if v v . Except for the requirement that the vibrational quantum number must... 

Schrodinger s eq.(5) looks pretty much the same as in standard BO theory as both schemes are based on a separability ansatz. They are, however, fundamentally different. While in the BO scheme the electronic wave functions On(p,R) are assumed to form complete orthonormal sets for all R-parameters (see, eg, refs [2, 18]), the present theory may put constraint as a function of the electronic state, so that not all nuclear configurations may be accessible. The new theory... 

Consider the Hartree-Fock (HF) ground state of the N-electron neutral cluster, One can form a complete orthonormal set of the (N — l)-electron basis functions, fI/v 11, applying the so-called physical excitation operators,... 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

The reference state of A-electron theory becomes a reference vacuum state 4>) in the field theory. A complete orthonormal set of spin-indexed orbital functions fip(x) is defined by eigenfunctions of a one-electron Hamiltonian Ti, with eigenvalues ep. The reference vacuum state corresponds to the ground state of a noninteracting A-electron system determined by this Hamiltonian. N occupied orbital functions (el < pi) are characterized by fermion creation operators a such that a] ) =0. Here pt is the chemical potential or Fermi level. A complementary orthogonal set of unoccupied orbital functions are characterized by destruction operators aa such that aa < >) = 0 for ea > p and a > N. A fermion quantum field is defined in this orbital basis by... 

Any basis set, consisting of a complete orthonormal set of functions, should produce the correct eigenvalues after variational minimization, e.g. 

Both the LCAO-MO and the unit-cell density are represented by an infinite sum over a complete orthonormal set. In crystallography the number of... 

The exact Schrodinger equation of motion, equation (1), may be equivalently Stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Yj(x, R) is expanded in terms of the complete orthonormal set of functions y>i(x R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the x (R)6... 

According to von Neumann (9), an abstract Hilbert space X is a linear space A = x having a binary product , which satisfies the conditions, Eqs. (1.8)—(1.10), and which further contains all its limit points in the norm x = 1/2 and is separable. The last assumption means that there exists an enumerable set Jf" = xk which is everywhere dense in and which ensures the existence of at least one complete orthonormal set

[Pg.99]

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