Variation principle constraints

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

Now that we have decided on the form of the wave function the next step is to use the variational principle in order to find the best Slater determinant, i. e., that one particular Osd which yields the lowest energy. The only flexibility in a Slater determinant is provided by the spin orbitals. In the Hartree-Fock approach the spin orbitals (Xi 1 are now varied under the constraint that they remain orthonormal such that the energy obtained from the corresponding Slater determinant is minimal... 

In order to connect this variational principle to density functional theory we perform the search defined in equation (4-13) in two separate steps first, we search over the subset of all the infinitely many antisymmetric wave functions Px that upon quadrature yield a particular density px (under the constraint that the density integrates to the correct number of electrons). The result of this search is the wave function vFxin that yields the lowest... 

The only term for which no explicit form can be given, i. e the big unknown, is of course Exc- Similarly to what we have done within the Hartree-Fock approximation, we now apply the variational principle and ask what condition must the orbitals cp fulfill in order to minimize this energy expression under the usual constraint of ((p I (pj) = ,j The resulting equations are (for a detailed derivation see Parr and Yang, 1989) ... 

Eq. (22) have been derived from the variation principle alone (given the structure of H) they contain only the single model approximation of Eq. (9) the typically chemical idea that the electronic structure of a complex many-electron system can be (quantitatively as well as qualitatively) understood in terms of the interactions among conceptually identifiable separate electron groups. In the discussion of the exact solutions of the Schrodinger equation for simple systems the operators which commute with the relevant H ( symmetries ) play a central role. We therefore devote the next section to an examination of the effect of symmetry constraints on the solutions of (22). 

It is found empirically and of course is predictable theoretically that, when using a model for molecular electronic structure, the set of eigenfunction equations associated with the operators commuting with H are constraints on the action of the variation principle if Et is computed from R subject to symmetry constraints and E2 is computed in the same model with no such constraints then (2)... 

As we stated at the outset, the variation principle is our starting point and the interpretation of valence and molecular electronic structure our aim the imposition of some of the more familiar constraints on the action of the variation principle deflects us from that aim. 

In examining numerical approximations it is as well to bear in mind the general qualitative conclusion of our brief examination of symmetry constraints. In broad terms the result was the simpler the model the more severe the effect of any constraint on the variation principle. This result cannot be carried over directly and used in numerical work since numerical approximation schemes can rarely be brought into a sufficiently coherent logical and mathematical form for analysis. Nevertheless it seems likely that this result can be used as a guideline — a rule of thumb . We therefore expect that the imposition of formal constraints and consistency requirements (derived from a higher level of approximation or the exact solution) on numerical approximation schemes is likely to have far-reaching consequences — particularly on the... 

We therefore conclude that attempts to impose constraints which are based on the formal properties of the exact Schrodinger equation leads to contradictory and even self-contradictory results besides placing unnecessary limitations on the action of the variation principle. That is not to advocate wilful inconsistency, of course. We shall insist on consistencies but within the confines of our orbital-basis variational model. 

It is clear from the preceding sections that the powerful A-representable constraints from the orbital representation do not extend to the spatial representation. This suggests reformulating the variational principle in g-density functional theory in the orbital representation. 

In one sense, research in theoretical chemistry at Queen s University at Kingston originated outside the Department of Chemistry when A. John Coleman came in 1960 as head of the Department of Mathematics. Coleman took up Charles Coulson s challenge150 to make the use of reduced density matrices (RDM) a viable approach to the N-electron problem. RDMs had been introduced earlier by Husimi (1940), Lowdin (1955), and McWeeny (1955). The great attraction was that their use could reduce the 4N space-spin coordinates of the wavefunctions in the variational principle to only 16 such coordinates. But for the RDMs to be of value, one must first solve the celebrated N-repre-sentability problem formulated by Coleman, namely, that the RDMs employed must be derivable from an N-electron wavefunction.151 This constraint has since been a topic of much research at Queen s University, in the Departments of Chemistry and Mathematics as well as elsewhere. A number of workshops and conferences about RDMs have been held, including one in honor of John Coleman in 1985.152 Two chemists, Hans Kummer [Ph.D. Swiss Federal Technical... 

Focusing on the variational principle present at the heart of the Density Functional formalism (actually a minimum principle), Eq.(4) must be minimised with respect to the variations of the wavefunctions, subject to the following orthonormalisation constraints ... 

The state-by-state CG algorithm of Teter, Payne and Allan can now be adapted to the variational principle Eq.(20), under constraints Eq.(22). By construction, such an algorithm will be unconditionnally stable, since the second-order derivative of the electronic energy is always decreased in the successive steps of the CG algorithm. It is presented in detail in Ref. [13], where its specific advantages are discussed. Now, let us turn to the specific features related to the treatment of an extended system of electrons and nuclei [1,6,13,14]. 

Stationary points of the functional [c] should be calculated through variation of the coefficients c. Kohn s variational principle requires the wave function on dS to remain fixed during the variation, 6fa = 0. In view of Eq. (27), this means that variation of the Ck is subject to the additional condition Y.k Tak Sck — 0. The standard way to solve a variational problem with constraints is to use undetermined Lagrange multipliers [234]. A technical realization of this method, which we do not describe here, is given in Ref. 60. Using it, one obtains a compact expression for a set of coefficients c which render [c] stationary, namely... 

Here, FHK is the HK functional. Both (14) and (15) are derived from the variational principles for the energy (in terms of the electron density and wave function, respectively), and both p, and E enter the equations as Lagrange multipliers associated with normalization constraints. The equal status of information carriers suggests that, just as for p(r), we can derive a variational equation for o-(r). 

The Schrodinger variational principle can be applied directly for fixed occupation numbers nt, E is required to be stationary subject to (i j) = 8y. To simplify the derivation, trial orbitals can be required to be orthogonal, but diagonal Lagrange multipliers e, are used for the normalization constraint. The variational condition is... 

With the normalization constraint, any ground-state wavefunction PV of Hv minimizes (1frl//vl1fr) according to the variational principle of quantum mechanics. Hence, any ground-state wavefunction, corresponding to a particular ground-state density pv, also minimizes this quantity when the variation is restricted to that density. In such a restricted variation the potential contribution (5) is constant, and it then follows that also ( P T + W F) is minimized, or... 

It should be emphasized, that this density functional for the ground-state energy differs from the (HK) functional for the energy, Ev[p, in which the external potential is fixed (not related to the trial density p). Only for the exact ground-state density Ev[p0] = E[p0. The variational principle of equation (16), also defined for the fixed v(r), determines the minimum of Ev[p subject to the density normalization constraint iV[p] = J p(r)dr = N. The associated Euler equation for the optimum (ground-state) density p(r) = p /V0. v r] = p0(r) reads ... 

This leads to the computationally useful variational principle minimizing iipk] = ( P //molecule I P) with respect to wave functions which satisfy the constraints from Eqs. (4) and (5) yields the exact ground-state energy. Performing this minimization by introducing a Lagrange multiplier for the normalization constraint of Eq. (4), one obtains the famous Schrodinger equation... 

In other words, the functional equivalence of Eq. (8) can be met just by requiring that D2 be TV-representable and this, in turn, means that one must determine the necessary and sufficient conditions for characterizing V% as a set containing TV-representable 2-matrices. This problem, however, is still unsolved. If not enough conditions are introduced in order to properly characterized V%, the minimum of Eq. (8) is not attained at E0 but at some other energy E 0 < E0. Thus, the upper-bound constraint of the quantum mechanical variational principle no longer applies and one can get variational" energies which are below the exact one[53]. 

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