Electronic Hilbert space

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

Then, two things (that are actually interdependent) happen (1) The field intensity F = 0, (2) There exists a unique gauge g(R) and, since F = 0, any apparent field in the Hamiltonian can be transformed away by introducing a new gauge. If, however, condition (1) does not hold, that is, the electronic Hilbert space is truncated, then F is in general not zero within the tmncated set. In this event, the fields A and F cannot be nullified by a new gauge and the resulting YM field is true and irremovable. 

From the A -electron Hilbert-space eigenvalue equation, Eq. (2), follows a hierarchy of p-electron reduced eigenvalue equations [13, 17, 18, 47] for 1 < p < N — 2. The pth equation of this hierarchy couples Dp,Dp+, and and can be expressed as... 

Let be Bm = electron function basis. Slater-determinants constructed over Bm span an orthonormal, jx = ( ) dimensional subspace of the N-electron Hilbert space. The projection of the exact wave function in this subspace ( ) can be given as a linear... 

Real space algorithms (section 4) allow for mappings between present day computer programs and strict molecular quantum mechanics [10,11]. It is the separability of base molecular states that permits characterizing molecular states in electronic Hilbert space and molecular species in real space. This feature eliminates one of the shortcomings of the standard BO scheme [6,7,12]. Confining and asymptotic GED states are introduced. In section 5 the concept of conformation states in electronic Hilbert space is qualitatively presented. 

The decomposition eq. (2-6) of the spin-free space FSP induces a decomposition on the Pauli-allowed portion of the Hilbert space of the Hamiltonian H of eq. (2-1). The Hamiltonian H which includes spin interactions may operate on any ket of the space Fsp V", where V is the electronic spin space. Here the symbol indicates a tensor product, so that Fsp Va consists of all spatial-spin kets which are composed of linear combinations of a simple product of a spatial ket of FSP and a spin ket of Va. The Pauli-allowed portion of the total A-electron Hilbert space of is... 

The electronic Coulomb interaction u(r 12) = greatly complicates the task of formulating and carrying out accurate computations of iV-electron wave functions and their physical properties. Variational methods using fixed basis functions can only with great difficulty include functions expressed in relative coordinates. Unless such functions are present in a variational basis, there is an irreconcilable conflict with Coulomb cusp conditions at the singular points ri2 - 0 [23, 196], No finite sum of product functions or Slater determinants can satisfy these conditions. Thus no practical restricted Hilbert space of variational trial functions has the correct structure of the true V-electron Hilbert space. The consequence is that the full effect of electronic interaction cannot be represented in simplified calculations. 

In this appendix we generalise the expressions of the diabatic quantities first introduced in Sec. 2 for the ideal case of an exact two-level problem to a more realistic description. In a normal situation, the Hamiltonian has an infinite number of eigenstates, and there is no finite number of strictly diabatic states [76] that can describe a given pair of adiabatic states [77-80]. Instead, one can define a unitary transformation of the adiabatic states generating two quasidiabatic states characterised by a residual non-adiabatic coupling, as small as possible, but never zero (see, e.g., [5,24,32-35]). In practice, the electronic Hilbert space is always truncated to a finite number of configurations. In what follows, we consider the case of MCSCF wavefunctions and make use of generalised crude adiabatic states adapted to this. 

When the external electric field is time-dependent, there is no well-defined energy of the molecular system in accordance with Eq. (100), and the wave function response can thus not be retrieved from a variational condition on the energy as in the analytic derivative approach described above. Instead the response parameters have to be determined from the time-dependent Schrodinger equation, a procedure which was illustrated in Section 3 for the exact state case. In approximate state theories, however, our wave function space only partially spans the 7V-electron Hilbert space, and the response functions that correspond to an approximate state wave function will clearly be separate from those of the exact state wave function. This fact is disregarded in the sum-over-states approach, and, apart from the computational aspect of slowly converging SOS expressions, it is of little concern when highly accurate wave function models are used. But for less flexible wave function models, the correct response functions should be used in the calculation of nonlinear optical properties. 

Consequently, within the spin free JV-electron Hilbert space, the MBPT for OSS states can be formulated as a SR theory with the spin free reference o), Eq-4, and the UGA representation of the perturbed and unperturbed Hamiltonians. The resulting theory has formally the same appearance as the spin orbital based MBPT theories for CS or HS OS cases. 

One question, however, cannot be avoided completely. It is again that of relativistic observables. In view of this interpretation, only an observable that leaves the subspace of positive energy invariant, is a good observable. With r/) e pos we should also require that Atp G pos, otherwise a measurement of the observable A would throw the state out of the electronic Hilbert space. 

These eigenfunctions define a basis in the N-electron Hilbert space. To account for the effects of the rest interaction V perturbatively, the full Hilbert space H of Hg is separated into two parts the model space M and its complement, TL M, which includes all the remaining dimensions. To these subspaces, we assign the projection operators... 

Fig. 10.6. Symbolic illustration of the principle of the Cl method with one Slater determinant o dominant in the ground state (this is a problem of the many electron wave functions so the picture caimot be understood literally). The purpose of this diagram is to emphasize a relatively small role of electronic correlation (more exactly, of what is known as the dynamical comlation i.e., correlation of electronic motion). The function jrci is a linear combination (the c coefficients) of the determinantal functions of different shapes in the marry-electron Hilbert space. The shaded regions correspond to the negative sign of the function the rxxlal surfaces of the added functions allow for the effective deformation of V o to have lower and lower average energy, (a) Since C is small in comparison to cq, the result of the addition of the first two terms is a slightly deformed V o. (b) Similarly, the additional excitations just make cosmetic changes in the function (although they may substantially affect the quantities calculated with it).

In practical applications of the theory, the computational problem is simplified by restricting the electronic Hilbert space to just two or three electronic states of interest, e.g. the ground state and two excited states. A further significant simpiification arises from symmetry selection rules. It follows from the definition (10) that the first-order intra-state coupling constants can be nonzero only for totally symmetric modes. Wnn (0) is zero if and transform according to different irreducible representations. The first-order inter-state coupling constant A is nonzero only for modes which transform according to the irreducible representation Tq which fulfils... 

The first step toward a practical relativistic many-electron theory in the molecular sciences is the investigation of the two-electron problem in an external field which we meet, for instance, in the helium atom. Salpeter and Bethe derived a relativistic equation for the two-electron bound-state problem [135,170-173] rooted in quantum electrod)mamics, which features two separate times for the two particles. If we assume, however, that an absolute time is a good approximation, we arrive at an equation first considered by Breit [101,174,175]. The Bethe-Salpeter equation as well as the Breit equation hold for a 16-component wave function. From a formal point of view, these 16 components arise when the two four-dimensional one-electron Hilbert spaces are joined by direct multiplication to yield the two-electron Hilbert space. 

The solution of the electronic Schrodinger equation, Eq. (8.76), requires the integration of a complicated partial differential equation depending on 3N electronic coordinates as variables. The complexity of this problem increases with the number of electrons in the molecule. In order to establish a general solution strategy it is mandatory to first study the underlying formal framework of many-particle quantum mechanics, namely that of a linear vector space — the Hilbert space (cf. section 4.1). The N-electron Hilbert space, which hosts the total quantum mechanical state vector, is then constructed by direct multiplication of the one-electron Hilbert spaces. 

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