Ensemble v-representability

We have seen that the Lieb functional Fh is differentiable at the set of E-V-densities in S and nowhere else. For this reason it is desirable to know a bit more about these densities. The question therefore is which densities are ensemble v-representable In this section we will prove a useful result which will enable us to put the Kohn-Sham approach on a rigorous basis. 

This statement means the following. Suppose we take an arbitrary density n0 from the set S, then for every e 0 we can find an E-V-density n such that ll 0 — n e and 0 — 3 e. In other words, for every density in the set S there is an E-V- 

Now the functional on the right hand side of the inequality sign is, for a given v, a linear functional of n. The inequality sign tells us that this functional lies below the graph of El[w]. A linear functional with this property is called FL-bounded. Let us give a general definition of these linear functionals. Let F be a functional F B — 1Z from a normed function space (a Banach space) B to the real numbers. Let B be the dual space of B, i.e., the set of continuous linear functionals on B. Then L E B is said to be / -bounded if there is a constant C such that for all n G B 

Theorem 12 (Bishop-Phelps I). Let F B — 1Z be a lower semicontinuous convex functional on a real Banach space B. The functional F can take the value +oo but not everywhere. Then the continuous tangent functionals to F are B -norm dense in the set of F-bounded functionals in B.  

This means that if Lq is some F-bounded functional then we can find a set of tangent functionals Lk such that 

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The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

We will denote densities n(r) of this type, which are obtained from an orthonormal set of ground states I i = 1- -q corresponding to a potential v, as ensemble v-representable densities, or for short E-V-densities. We further denote the set of all E-V-densities generated by potentials in L,/2 + L00 as B. A density will be called a pure state v-representable density or for short PS-V-density if it can be written as (r) = (1frlu(r)l1fr), where I P) is a ground state. Obviously PS-V-densities are special cases of E-V-densities and the set of PS-V-densities is therefore a subset of the set of E-V-densities. 

But we also know that FEHK is convex on the set of E-V-densities. This leads to a contradiction and hence we must conclude that n cannot be a pure state density of any potential. The density h is, however, a convex combination of ground state densities corresponding to the same external potential and therefore, by definition, an ensemble v-representable density. We therefore have constructed an E-V-density, which is not a PS-V-density. For an explicit numerical example of such a density we refer to the work of Aryasetiawan and Stott [18],... 

We therefore now have established that FL is a convex functional on a convex space. This is important information which enables us to derive the Gateaux differentiability of the functional FL at the set B of ensemble v-representable densities. We will discuss this feature of FL in the next section. 

We see that this is simply the Lieb functional with the two-particle interaction omitted. All the properties of the functional Fh carry directly over to Th. The reason is that all these properties were derived on the basis of the variational principle in which we only required that 7 + IV is an operator that is bounded from below. This is, however, still true if we omit the Coulomb repulsion W. We therefore conclude that Th is a convex lower semicontinuous functional which is differentiable for any density n that is ensemble v-representable for the noninteracting system and nowhere else. We refer to such densities as noninteracting E-V-densities and denote the set of all noninteracting E-V-densities by >0. Let us collect all the results for 7) in a single theorem ... 

The differentiability of different functionals used in density-functional theory (DFT) is investigated, and it is shown that the so-called Levy Lieb functional FLL[p] and the Lieb functional FL[p] are Gateaux differentiable at pure-state v-representable and ensemble v-representable densities, respectively. The conditions for the Frechet differentiability of these functionals are also discussed. The Gateaux differentiability of the Lieb functional has been demonstrated by Englisch and Englisch (Phys. Stat. Solidi 123, 711 and 124, 373 (1984)), hut the differentiability of the Levy-Lieb functional has not been shown before. 

In the degenerate case we can have a situation, where a linear combination of ground-state densities is not necessarily itself a ground-state density. This has the consequence that the HK [equation (11)] and the Levy-Lieb [equation (38)] functionals are not necessarily convex, which for many applications is a disadvantage. A convex functional can be constructed by considering ensemble-v-representable (E-v-representable) densities [3,12,11]... 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

The first relation follows from the fact that if the density n is a pure state v-representable density then the minimizing density matrix for FL is a pure state density matrix. The second relation also easily follows. We take n to be an E-V-density which is not a PS-V-density. There is a ground state ensemble density matrix D[n for which we have... 

This shows that the functional FL[p] is Gateaux differentiable at any density generated by an ensemble of ground-state wave functions, i.e., at any Zs-v-representable density,... 

Such an ensemble generalized ground-state energy functional, E = E[N, v] = E[p[N. i l- represents the thermodynamic potential of the N, v -representation, with the corresponding generalized Hellmann-Feynman expression for its differential (see equations (17), (22) and (27)) ... 

Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...

The partial molar volume, which is a very important quantity to probe the response of the free energy (or stability) of protein to pressure, including the so-called pressure denaturation, is not a canonical thermodynamic quantity for the (V, T) ensemble, since volume is an independent thermodynamic variable of the ensemble. The partial molar volume of protein at infinite dilution can be calculated from the Kirkwood-Buff equation [20] generalized to the site-site representation of liquid and solutions [21,22],... 

Figure 16.8 Model GCN4-p1 peptides, (a) Helical wheel diagram and sequence of GCN4-pl analogue. C = acetamidocysteine. (b) Structures oftrifluoroleucine (L) and trifluorovaline (V) used to stabilize peptide ensembles. The asterisk indicates unresolved stereochemistry, (c) Model structure of GCN4-p1 (PDB code 2ZTA). Side-chains of V and L residues at a and d positions are shown as spheres. Side-chains of Asn residues are shown in stick representation. The structure was generated using MacPyMOL (DeLano Scientific LLC, Palo Alto, CA, U.S.A.).

Figure 13 Structures of PTPs include two important motifs, the P-loop that bears the cysteine nucleophile within the general signature motif (H/V)Cp<)5R(S/T), and the WPD-loop, which includes an important aspartic acid, a general acid-base catalyst. Substrate binding by the P-loop promotes a change of the WPD-loop conformation from an open, inactive to a closed, active conformation in which the aspartic acid completes the catalytic ensemble used for catalysis. The representation in this figure was created using PyMol from the PTP1B structures in apo-bound (PDB 2CM2) and inhibitor-bound (PDB 1BZJ) forms.

A, the Helmholtz energy, is a thermodynamic potential for the canonical ensemble. Qnvt, often symbolized simply as Q, is the canonical partition function. The last of Eqs. (46) defines a fundamental equation in the Helmholtz energy representation by expressing as a function of N, V, T. Often in classical systems it is possible to separate the energy contributions that depend on the momenta only (kinetic energy K) from the potential energy V, which depends only on the coordinates. When Cartesian coordinates are used as degrees of freedom, for example, the partition function can be factorized as ... 

The functional F n is defined via (1.55) for aU densities n(r) which are JV-representable , i.e., come from an antisymmetric AT-electron wave-function. We shall discuss the extension from wavefunctions to ensembles in Sect. 1.4.5. The functional derivative SF/6n r) is defined via (1.58) for all densities which are w-representable , i.e., come from antisymmetric AT-electron ground-state wavefunctions for some choice of external potential v(r). 

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