Slater condition

A careful reader will observe that this algebraic transformation will produce a dual SDP problem that does not have y G IR such that the matrix in Eq. (16) has all of its eigenvalues positive and, therefore, will not satisfy the Slater conditions. However, numerical experiments have shown that practical algorithms stUl can solve these problems efhciently [16]. 

More abstractly the condition Tr (p+p ) = iV implies that the part of the Hilbert space defined by the projection operator p should be fully contained in the part defined by the projection operator p+. If we now vary p slightly so that this condition is no longer fulfilled, Eq. II.GO shows that the pure spin state previously described by the Slater determinant becomes mixed up with states of higher quantum numbers S = m+1,. . . . The idea of the electron pairing in doubly occupied orbitals is therefore essential in the Hartree-Fock scheme in order to secure that the Slater determinant really represents a pure spin state. This means, however, that, in the calculation of the best spin orbitals y>k(x), there is a new auxiliary condition of the form... 

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

Here we derive the conditions of orbital phase for the cyclic orbital interactions. The A B delocalization is expressed by the interaction between the ground configuration C Q and the electron-transferred configuration tBp(A B) (Scheme 3). A pair of electrons occupies each bonding orbital in which is expressed by a single Slater determinant 0 ... 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

The calculation of the cross matrix elements (6) is somewhat more difficult, because the Slater Determinants involved in them are constructed with two sets of non-orthonormal spinorbitals. This calculation, however, may be greatly simplified, if the two sets are assumed to be corresponding, that is, if they fulfill the following condition [14] ... 

Corongui, F.P., Poll, G., Diansani, M.U., Cheeseman, K.H. and Slater, T.F. (1986). Lipid peroxidation and molecular damage to polyunsaturated fatty acids in rat liver. Recognition of two classes of hydroperoxides formed under conditions in vivo. Chem. Biol. Interactions 59, 147-155. 

In the present work, it was desired to 1) verify the prediction that the Henry s law constant controlled variations in the experimental volatilization rate constants under constant environmental conditions, and 2) compare experimental volatilization rate constants to predicted constants using the two methods for estimating kg for water and the molecular weight adjustment procedure of Liss and Slater as discussed above. 

Although they did not obtain a closed-form analytic expression for the three-dimensional case, they dealt with a trasformed one-matrix for the single Slater determinant constructed from plane waves, and rewrote the energy in terms of this transformed matrix. The conditions on the transformation were not imposed through the Jacobian but rather through the equations ... 

Note that these orbitals satisfy the equidensity condition iA (a, b, c) b, c) = p(a, b, c)/N hence, a single Slater determinant constructed from these orbitals yields the density ... 

In addition, since the non-local part of the 1-matrix for the single Slater determinant (for a non-interacting system) also appears in Eqs. (159) and (160), it is necessaiy that we select a generating function (p n xs) a particular orbit This wavefunction satisfies the condition of minim-... 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

C. Necessary Conditions from the Slater Hull (R, R) Conditions... 

The similarities and differences between methods based on the electron density, electron-pair density, and reduced density matrices have recently been reviewed [5]. This chapter is not intended as a comprehensive review, but as a focused consideration of A-representability constraints that are applicable to diagonal elements of reduced density matrices. Such constraints are useful both to researchers working with the Q-density and to researchers working with g-electron reduced density matrices, and so we shall attempt to review these constraints in a way that is accessible to both audiences. Our focus is on inequalities that arise from the Slater hull because the Slater hull provides an exhaustive list of A-representability conditions for the diagonal elements of the density matrix, Although the Slater hull constraints... 

Henceforth, we will use the statements Vq is in the Slater hull, Tg satisfies the Slater hull conditions, and Tg satisfies the Q, K) conditions interchangeably. 

Necessary conditions based on the Slater hull have been pursued by Kummer, McRae and Davidson, Yoseloff and Kuhn, and Deza and Laurent, among others [24-26, 29, 42-44]. 

Even for a relatively small system, obtaining the Q, R) conditions is computationally challenging [25]. For example, using the 2-matrix to describe the beryllium atom in a minimal basis would require the (2,10) conditions for a four-electron system. In this case, the Slater hull is a polyhedron with on the order of ten billion facets. Only small R is interesting for computational applications. 

As previously discussed, when developing necessary conditions for N-representability, it is useful to consider the subset of Hamiltonians whose ground-state energy is zero. Applying this idea to the Slater hull, the following Hamiltonians arise as important constraints ... 

In fact, even the full set of (R, K) conditions is insufficient. Since the full set of Slater hull constraints is not sufficient to ensure the A-representability of the /f-matrix, there exist some F that satisfy the Slater hull constraints,... 

The Slater hull constraints represent the entire family of A -representabUity constraints that can be expressed using only the diagonal elements of the reduced density matrix [25, 43]. That is, the complete set of (g, K) conditions is necessary and sufficient to ensure the A -representability of the g-density. 

Although every g-density in the Slater hull is A-representable, not every g-matrix that satisfies the Slater hull constraints is A-representable. That is, if Pii...iQ =, g ,j satisfies the (g, K) conditions, some choices for the off-... 

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