Exclusion principle violating

Here, the first term represents the correlation energy while the second one stands for the exclusion principle violating (EPV) terms which involve a forbidden double excitation over ( ),) i.e., those fulfilling G ( )i) = 0). The third term excludes the effect of all higher than double excitations which are already included in the determinantal space Sm (they should not be considered when describing in A the effect of the outer space). Then, we may write equation (12) as. 

There is a price to pay for the separability, or equivalently for the presence of only connected diagrams. Somewhat like in traditional many-body theory, one must be ready to accept so-caUed EPV (exclusion-principle violating) cumulants. Typical EPV cumulants are nonvanishing kk for k> n, while yj = 0 for k > n. 

It can be shown [95] that in such a case the T4 contribution cancels the exclusion principle violating (EPV) quadratic terms [59]. This realization led us to the formulation of the so-called ACPQ method (CCSD with an approximate account for quadruples) [95], as well as to CCDQ and CCSDQ [86], the latter also accounting for singles. Up to a numerical factor of 9 for one term involving triplet-coupled pp-hh t2 amplitudes [95], this approach is identical with an earlier introduced CCSD-D(4,5) approach [59] and an independently developed ACCD method of Dykstra et al. [96,97]. This method arises from CCSD by simply discarding the computationally most demanding (i.e., nonfactorizable) terms (see Refs. [59,95-97] for details). 

For the sake of correctness, it is necessary to note that we disregarded such theoretically important concepts as the linked cluster (linked connected graph) theorem and the exclusion principle violating (EPV) diagrams. This is in accordance with our aim to maintain the practical nature of this review. The linked cluster theorem and EPV diagrams are of importance in the fourth and higher orders of the perturbation theory which, in our opinion, shall hardly be accessible to routine calculations in the foreseeable future. For detailed information on the linked cluster theorem and EPV diagrams see Refs.9,33,34,4a ... 

Table 4 Ground state Heisenberg energies per site for undistorted polymers (see Refs. 22, 23, 34, 35, and 45 and references therein for other available results). EPV stands for Exclusion Principle Violating infinite summation perturbation theory (see Ref. 34 for more details).

The sum over internal lines is unrestricted so (linked) exclusion principle violating (e-v38 or EPV27) terms are included in the sums. Note that either restricting sums to eliminate both unlinked and linked EPV diagrams or unrestricting summations and doing a simple cancellation of unlinked terms satisfies the Pauli exclusion principle.27... 

Only much later were we able to account for some of these difficulties [109-112] by developing methods that, under certain conditions, account for the T4 clusters. In cases when the projected UHF method provides the exact pair clusters, it can be shown [109] that the T4 contribution cancels the contribution from the non-linear terms representing the important exclusion-principle-violating (EPV) diagrams. Except for a numerical factor associated with the triplet-coupled pp-hh t2-amplitudes, this method—referred to as ACPQ or CCDQ —is identical to the ACCD approach that was developed independently by Dykstra s group [113-116]... 

Here, the first term represents the correlation energy while the second one stands for the exclusion principle violating (EPV) terms which involve a forbidden double excitation over (j),) (i.e., those fulfilling = 0). The third term... 

The natural energy decomposition analysis (the keyword is NEDA) of Glendening and Streitwieser provides a more comprehensive picture of the various energy components contributing to intermolecular interactions. The NEDA decomposition mimics in some ways the older Kitaura-Morokuma analysis, but it avoids the use of non-orthogonal (and exclusion principle-violating) wavefiinctions for the two monomers, with the attendant interpretational ambiguities. 

This result evidently arises from a purely algebraic identity, and the fact that some of the diagrams included in the summation may be exclusion-principle violating diagrams (EPVDs), when the same index appears more than once at some given level, is of no consequence. 

We recall the approximation that leads to the single-reference CEPA (SR-CEPA) [42]. CEPA(O) emerges when the SRCC equations are totally linearized, and the virtual space is restricted to functions reached via the hamiltonian. In the same spirit, we can generate the analogous SS-MRCEPA(O), if we hnearize our SS-MRCC equations and retain only the one- and the two-body cluster operators in T. In the SR-CEPA(2), the diagonal exclusion principle violating (EPV) terms are additionally retained, and we propose a similar scheme, SS-MRCEPA(2), where analogous terms in H are retained in the SS-MRCC theory. 

Again, for the filled orbitals L = 0 and 5 = 0, so we have to consider only the 2p electrons. Since n = 2 and f = 1 for both electrons the Pauli exclusion principle is in danger of being violated unless the two electrons have different values of either or m. For non-equivalent electrons we do not have to consider the values of these two quantum numbers because, as either n or f is different for the electrons, there is no danger of violation. 

Primary steric effects are due to repulsions between electrons in valence orbitals on atoms which are not bonded to each other. They are believed to result from the interpenetration of occupied orbitals on one atom by electrons on the other resulting in a violation of the Pauli exclusion principle. All steric interactions raise the energy of the system in which they occur. In terms of their effect on chemical reactivity, they may either decrease or increase a rate or equilibrium constant depending on whether steric interactions are greater in the reactant or in the product (equilibria) or transition state (rate). 

Even if the g-density is A -representable, this Q-matrix is not A -representable because its largest eigenvalue exceeds the upper bound N /Q N — Q+ ). (That is, this g-matrix violates the Pauli exclusion principle for g-tuples of electrons.) Approximating the correction term, Tp Jpg[ (]], seems difficult, and neglecting this term would give poor results, although the results improve with increasing Q [2, 10]. 

A common reference density, first used by Roux and Daudel (1955), is the superposition of spherical ground-state atoms, centered at the nuclear positions. It is referred to as the promolecule density, or simply the promolecule, as it represents the ensemble of randomly oriented, independent atoms prior to interatomic bonding. It is a hypothetical entity that violates the Pauli exclusion principle. Nevertheless, the promolecule is electrostatically binding if only the electrostatic interactions would exist, the promolecule would be stable (Hirshfeld and Rzotkiewicz 1974). The difference density calculated with the promolecule reference state is commonly called the deformation density, or the standard deformation density. It is the difference between the total density and the density corresponding to the sum of the spherical ground-state atoms located at the positions R... 

The promolecule density shows (3, — 1) critical points along the bond paths, just like the molecule density. But, as the promolecule is hypothetical and violates the exclusion principle, it would be incorrect to infer that the atoms in the promolecule are chemically bonded. In a series of topological analyses, Stewart (1991) has compared the model densities and promolecule densities of urea,... 

Using the above definitions for the four quantum numbers, we can list what combinations of quantum numbers are possible. A basic rule when working with quantum numbers is that no two electrons in the same atom can have an identical set of quantum numbers. This rule is known as the Pauli Exclusion Principle named after Wolfgang Pauli (1900-1958). For example, when n = 1,1 and mj can be only 0 and m can be + / or -1/ This means the K shell can hold a maximum of two electrons. The two electrons would have quantum numbers of 1,0,0, + / and 1,0,0,- /, respectively. We see that the opposite spin of the two electrons in the K orbital means the electrons do not violate the Pauli Exclusion Principle. Possible values for quantum numbers and the maximum number of electrons each orbital can hold are given in Table 4.3 and shown in Figure 4.7. 

In addition to the Coulombic forces, there is a repulsive force which operates at short distances between ions as a result of the overlapping of filled orbitals, potentially a violation of the Pauli exclusion principle. This repulsive force may be represented by the equation ... 

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