Variational Approaches to the N-representability Problem

Transcription

1 Variational Approaches to the -representability Problem Paul W. Ayers Department of Chemistry; McMaster University I consider it useless and tedious to represent what exists, because nothing that exists satisfies me. ature is ugly, and I prefer the monsters of my fancy to what is positively trivial. Charles Baudelaire Salon of Curiosités Esthétiques (1868) 1

2 I. Motivation Density Functional Theory isn t really a black box: Problems with systems with strong static correlation. (E.g., systems where the single determinantal reference is poor.) Problems with systems where long-range correlation is important (i.e., the exchange-correlation hole is not very localized.) (E.g., van der Waals forces, multi-center bonding, associative transition states,.....) Semi-ab initio functionals can fail when describing unconventional chemistry, e.g., highly charged systems. In general, existing functionals are not systematically improvable. Good ews: Existing Density-Functionals are quite accurate for thermodynamic properties and (in most cases) chemical dynamics. Good ews: When DFT works, the accuracy/computational cost ratio of density-functional theory far exceeds that of most other methods. Can we improve DFT without using wave-function-based methods?

3 Past the density, but not yet to the wave function.... ( rr, ) γ Γ ( r, r ; r, r ) 1 1 Γ ( r, r, r; r, r, r ) Γ 4 full-ci ρ ( r) ( rr) ρ, ρ3( r1, r, r3) ρ 4( r1, r4) ρ =Ψ Ψ HF,CIS CCSD, etc. CCSDT, etc. CCSDTQ, etc. full-ci Baerends,Buijse,Cioslowski,Coleman,Davidson,Donnelly,Garrod,Goedecker Levy,Mazziotti,Parr,Percus,Umrigar,Valdemoro... Davidson,Furche,Levy,agy,Pistol,Samvelyan,Weinhold,Wilson,Ziesche... classic quantum chemistry Polydensity alternative: Gori-Giorgi,Percus,Savin. 3

4 Outline of the Remainder of the Talk II. The -representability Problem III. The -representability problem: Special Case; electron pair density. IV. Variational approaches to the -representability problem: Special Case; electron pair density. V. Variational Approaches: General Case. VI. Specific Variational Approaches: Density Matrices, etc.. VII. Algorithmic Considerations VIII. The -representability Problem, Revisited 4

5 II. The -representability Problem Given: A descriptor, f ( τ ), which contains enough information to describe a molecular electronic system. There exists: An energy function, E [ f ], that depends only v, on the descriptor and the identity of the system (as encapsulated by the external potential, v ( r ), and the number of electrons, ). There exists: A variational principle for the ground-state energy, namely: E v ; = min E f [ ] v[ ] gs.., -representable f ( τ) 5

6 The variational principle follows directly from the variational principle for the wave function: Ψ Hˆ v, Ψ Egs..[ v; ] = min ΨΨ all appropriately antisymmetric -fermion wavefunctions or density matrix E..[ v; ] = min Tr ˆ Hˆ, Γ g s v all -fermion density matrices This means that: E v ; = min E f [ ] v[ ] gs.., all f ( τ) that correspond to a system containing fermions If ( ) f τ corresponds to a system of fermions, then it is said to be -representable. 6

7 Definition: The descriptor, f ( τ ), is said to be - representable if and only if it corresponds to a system of -fermions. Therefore, if f ( τ ) is - representable, then there exists some fermionic -electron density matrix, Γ, that is consistent with f ( τ ). I.e., f τ is -representable Γ Γ yields f τ ( ) ( ) ( ) otation: denotes the set of -representable f ( τ ). The -representability problem: Find a way to constrain the variational principle so that the correct groundstate energy is obtained: min E f E v; = min E f [ ] [ ] f ( τ) [ ] v, gs.. v, f ( τ) 7

8 All... popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that. Terry Eagleton Against the Grain 8

9 III. Variational Approaches to the -representability problem: Special Case; electron pair density. The pair density is the probability of observing an electron at x 1 and x. It is related to the structure factor in X-ray/neutron scattering. ( x, x ) ( ri x ) ( rj x ) ρ Ψ δ δ Ψ 1 1 i= 1 j i = Tr δ ( r x1) δ ( r x) Γ i= 1 j i i j Simple Properties of the pair density: normalization nonnegativity symmetry ( 1 ) ρ (, ) ρ = x x dxdx 1 1 ( ) 0 ρ x, x 1 ( x, x ) = ρ ( x, x ) 1 1 9

10 Hohenberg-Kohn-Ziesche Theorem: ρ all observable properties, including the energy and its components. 1 ρ x, x V ne Vee [ ] ρ = ( ) dxdx 1 1 x1 x [ ρ ] ρ ( x, x ) ( x ) + v( x ) 1 v = d d x x The Kinetic energy functional is not known exactly. Approximations are available. (Furche, Levy, March, agy,... * ) Variational Principle for the pair density: E v, = min T ρ + V ρ + V ρ [ ] [ ] [ ] [ ] gs.. ne ee -representable ρ -representability problem: We must restrict the variational principle to ρ( x1, x ) that correspond to -fermion systems. (If one fails to do this, there will always exist -body Hamiltonians for which the error from the variational calculation is arbitrarily large.) * J. Math. Phys. 46, (005); Ayers & Levy Chem. Phys. Lett. 415, 11 (005). 10

11 The assical -body Structure Problem Given: A potential representing the interaction between any V x, x. pair of particles inside a system, ( ) 1 assical -body structure problem: Find the best (lowestenergy) configuration of -classical particles interacting with this potential. early [ ] min ( x, x ) E V V i j x i= 1 j i i [ ] Ψ ( r, r ) Ψ. E V V i j i= 1 j i so: For every -representable ρ( x1, x) and any ( 1, ) E [ V] ρ ( x, x ) V( x, x ) dxdx This is the only -representability constraint on ρ. V x x, 11

12 For every -representable ρ( x1, x) and any ( 1, ) E [ V] ρ ( x, x ) V( x, x ) dxdx V x x, In fact, as long as V ( x1, x ) is continuous, [ ] inf ρ (, ) (, ) because: E V V d d = x x x x x x. -representable ρ [ ] = min ( rr, ) E V V i j xi i= 1 j i = xi inf V ( ri, rj) xi= xj= xk is i= 1 j i never true = inf Tr V ( rr, ) Γ Γ i= 1 j i i j min rj rj 3 Ψ=. + ε 0 j = 1 ε The solution usually looks like lim A ( ε π) e σ() j 1

13 For every -representable ρ( x1, x) and any V ( x1, x ), E [ V] ρ ( x, x ) V( x, x ) dxdx In fact, as long as V ( x1, x ) is continuous, [ ] inf ρ (, ) (, ) -representable ρ E V V d d = x x x x x x THEOREM: For any ( 1, ) there exists a ( 1, ) [ ] ρ (, ) (, ) ρ x x that is not -representable, V x x such that E V V d d > x x x x x x Known: The set of -representable ρ is a convex set. (This follows directly from the definition, ρ( x1, x) = Tr δ ( r 1) ( ) Tr ˆ i x δ rj x Γ = L ρ Γ i= 1 j i ( a) ( b) If ρ and ρ are -representable, then convex sums are also: ( a) ( b) ( ) ( ) ˆ a ( b) tρ + 1 t ρ = Tr Lρ { tγ ( 1 ) } + t Γ 13

14 Hahn-Banach Separation Theorem: Given two disjoint convex sets, S 1 and S, there exists a hyperplane that separates the sets. For sets of pair densities: (, ) w(, ) ρ r r r r drdr Q ρk S Q ρk S If the distance between S 1 and S is greater than zero, then the and can be replaced by strict inequalities. ρ Choose the first convex set to be the set of -representable pair densities, ; choose the second convex set to be a non-representable pair density, 1 14

15 For every -representable ρ( x1, x) and any ( 1, ) E [ V] ρ( x1, x) V( x1, x) dx1dx In fact, as long as V ( x1, x ) is continuous, E [ V] = inf ρ ( x, x ) V( x, x ) dxdx. -representable ρ V x x, THEOREM: For any ( 1, ) there exists a ( 1, ) [ ] ρ (, ) (, ) ρ x x that is not -representable, V x x such that E V > x1 x V x1 x dx1dx Proof: If ρ( x1, x) is not -representable, then it follows from the Hahn-Banach separation theorem that there exists a potential for which this is true. ρ x, x is -representable if and only if This means that ( 1 ) [ ] ρ (, ) (, ) for every possible V ( x, x ). E V x x V x x dxdx

16 Consequences -representable pair densities are nonnegative: ρ x, x is negative in the region Ω. Choose Suppose that ( ) 1 0 ( r1, r) Ω w Ω ( r1, r) = 1 ( r1, r) Ω. Then E [ w ] = 0. But ( ) w( ) ormalization: w + x, x = 1 Choose ( ) 1 1 ρ x1, x x1, x dx1dx < 0 ; then ( ) d d E [ w ] ρ x1, x x1 x + 1 = 1= Choose ( x x ) w 1 1, = 1; then ( ) d d E [ w ] i= 1 j i ρ x1, x x1 x + 1 = 1= i= 1 j i x1 x x1 x. So ( 1 ) ρ (, ) d d ( 1)!! ( )!! ( ) 16

17 Generalized Davidson Constraint: Choosing: then w ( r, r ) f ( r) f ( r ) = ( 1) + f ( r) ( 1) f r E [ w] = min f ( r i) xi i= 1 0 This implies that ρ r, r f r f r drdr f r ρ r dr ( ) ( ) ( ) ( ) ( ) ( ) There are other similar arguments for all other previously known -representability constraints on the pair density. 17

18 It is the last lesson of modern science, that the highest simplicity of structure is produced, not by few elements, but by the highest complexity. Ralph Waldo Emerson 18

19 For every -representable ρ( x1, x) and any ( 1, ) E [ V] ρ( x1, x) V( x1, x) dx1dx THEOREM: The pair density (, ) V x x, ρ x1 x is -representable, if and only if E V ρ x, x V x, x dxdx [ ] ( ) ( ) for every V ( x, x ) Here the classical -body energy is defined by [ ] min ( x, x ) E V V i j x i= 1 j i i This is not a practical solution because it requires us to solve every possible classical many-body problem. This is very hard. 19

20 IV. Variational approaches to the -representability problem: Special Case; electron pair density. THEOREM: The pair density (, ) Thus: ρ x1 x is -representable, if and only if E V ρ x, x V x, x dxdx [ ] ( ) ( ) for every V ( x, x ) [, ] = min [ ρ ] + [ ρ ] + [ ρ ] E v T V V gs.. ne ee -representable ρ = + + min T[ ρ] Vne [ ρ] Vee [ ρ] { ρ w( r1, r), ρw E [ ]} w This is not very practical; but if we constrained this result using only one (, ) w r r 1, then that would be more acceptable. Then: E [ v, ; w] min T + V + V { ρ ρw E [ ]} w [ ρ ] [ ρ ] [ ρ ] gs.. ne ee 0

21 So we have a lower bound: Egs..[ v, ; w] min T[ ρ] + Vne[ ρ] + Vee[ ρ] { ρ ρw E [ ]} w We would like for this lower bound to be as tight as possible. This suggests that we maximize over all the potentials, obtaining the tightest possible lower bound for a simple constrained variational principle. Theorem: [ ] E v, max min T + V + V [ ρ ] [ ρ ] [ ρ ] gs.. ne ee w w E w { ρ ρ [ ]} This construction produces the exact ground-state energy. That is, Egs..[ v, ] = max min T[ ρ] + Vne[ ρ] + Vee[ ρ] w ρ ρ w E [ w] { } 1

22 Theorem: This construction produces the exact ground-state energy. That is, Egs..[ v, ] = max min T[ ρ] + Vne[ ρ] + Vee[ ρ] w ρ ρ w E [ w] { } Known: The energy is a convex functional of the pair density. That is, for any two pair densities, we have that ( a) ( b) ( a) ( b) E v, tρ ( 1 t) ρ t E v, ρ ( 1 t) E v, ρ + + Implication: This means that the set of pair densities whose energy is too small is convex. Let L denote the set of density matrices with energy lower than the true ground-state energy, L = ρ E ρ E v {, [ ]..[, ]} v g s ( a) ( b) L is convex because if ρ and ρ are both in L, then ( a) ( b) ( a) ( b) E v, tρ ( 1 t) ρ t E v, ρ ( 1 t) E v, ρ + + Egs..[ v, ] ( a) ( b) Which implies that tρ + 1 t ρ L. ( )

23 Theorem: The exact ground-state energy is obtained by the max-min prob: Egs..[ v, ] = max min T[ ρ] + Vne[ ρ] + Vee[ ρ] w { ρ ρ w E [ ]} w Known: The set of pair densities with too low energy is convex. Known: The set of -representable pair densities is convex. Known: These two sets do not intersect because every -representable pair density has energy greater than or equal to the true energy. Implication: There exists some potential, w( x1, x ), that separates these w x, x such that sets. I.e., there exists a ( 1 ) (, ) (, ) [ ] > ρ (, ) (, ) ρ x1 x w x1 x dx1dx E w x1 x w x1 x dx1dx ρ ρ L 3

24 Theorem: The exact ground-state energy is obtained by the max-min prob: Egs..[ v, ] = max min T[ ρ] + Vne[ ρ] + Vee[ ρ] w { ρ ρ w E [ ]} w 4

25 V. Variational Approaches: General Case. Given: A descriptor that determines all properties of any molecular system. It is assumed that this is a reduced descriptor (i.e., less complex than the -electron wavefunction) and that it is a linear functional of the -electron density matrix. f ( τ ) = Tr Lˆ f Γ ( r,, r ) ( r,, r ) Γ = w Ψ Ψ * i 1 1 i 0 w 1; w = 1 Implication: Because f ( τ ) Lˆ ( ) Given: i i = Tr f Γ, the set of -representable f τ is a closed, convex set. It is assumed that some portion of the energy can be evaluated exactly in terms of this descriptor. This portion of the energy is denoted W = h ˆ f { ) i 5

26 Theorem: The remainder of the energy, F[ f ], can be exactly represented by a convex functional. Proof: Let F[ f ] denote the Legendre transform functional. I.e.: [ ] = sup min Tr ˆ, { ˆ vγ ) F f H h f Γ hˆ 1. This functional is exact. For a specific choice, ĥ 0, [ ] ˆ, { ˆ F f mintr H vγ h0 f ) Γ F f + hˆ ˆ 0 f min Tr H v, Γ = Egs.. v, [ ] { ) [ ] Γ If ĥ 0 is associated with a maximum, then the energy is exact. Otherwise one has the variational principle. 6

27 . This functional is convex. ( ) { ) ( ) ( ) ( ) F tf ( 1 t) f sup ˆ mintr Hˆ + = v, Γ h tf + ( 1 t) f ˆ Γ a b a b h ( ( )) t+ 1 t mintr Hˆ v, Γ Γ = sup ˆ ˆ ( a) ( ) { ) ( 1 ) ˆ b h t h f t { h f ) { ) ( ) ˆ ˆ a tmintr Hv, Γ t h f Γ = sup ˆ ( 1 ) mintr ˆ, ( 1 ) ˆ h t H vγ t h f Γ sup t min Tr Hˆ ˆ Γ h hˆ ( a) v, ( 1 ) Γ ( b) ˆ ( a) { ) t h f ( b) { ) ˆ ( b) { ) + sup ( 1 t ) min Tr Hˆ v, Γ ( 1 t) h f Γ t F f + t F f 7

28 Theorem: f ( τ ) is -representable if and only if { ˆ ) partial hf E ˆ h Here partial E ˆ ˆ ˆ h h L = fγ Proof: If f ( τ ) is -representable, then clearly If ( ) min { Tr ) { ) min { Tr ˆ ) Γ ˆ ˆ partial hf h L ˆ fγ = E h Γ f τ is not -representable then because is a convex set, we can use the Hahn-Banach separation theorem to obtain the proof. Theorem: The exact ground-state energy can be obtained using E v, = max min F f + hˆ f gs.. [ ] [ ] ˆ ˆ partial h hf E ˆ h 8

29 The -electron reduced density matrix * (, ;, ) ( 1 ) (,,, ) (, ) Γ r r r r = Ψ r r r r Ψ r r dr r Variational Principle: E = T Γ + V Γ + V Γ min [ ] [ ] [ ] gs.. ne ee Γ from antisymmetric Ψ -representability problem: We must restrict the variational principle to Γ that correspond to antisymmetric wavefunctions. * (If one fails to do this, there will always exist -body Hamiltonians for which the error from the variational calculation is arbitrarily large.) E = T Γ + V Γ + V Γ min [ ] [ ] [ ] gs.. ne ee Γ from antisymmetric Ψ * In practice, it is more convenient to consider any pair density that corresponds to an ensemble average of fermionic wave functions. 9

30 the only density matrices that cause problems are those that give too small an energy for some Hamiltonian. non--representable density matrices with energies that are too high could be ignored. E = T Γ + V Γ + V Γ min [ ] [ ] [ ] gs.. ne ee { Γ E[ Γ ] E ˆ g. s. for every H} This condition is actually identical to the -representability condition. That is, Γ is (ensemble) -representable if and only if [ ] [ ] [ ] T Γ + Vne Γ + Vee Γ Eg. s. for every Hamiltonian. Equivalently, if Γ is not -representable, there exists some system with T[ Γ ] + Vne [ Γ ] + Vee [ Γ ] < Eg. s. One can make the error arbitrarily large by scaling the Hamiltonian. E = T Γ + V Γ + V Γ min [ ] [ ] [ ] gs.. ne ee { Γ E[ Γ ] E ˆ g. s. for every H} Actually, we can get the right energy if we only ensure that the energy is greater than the ground-state energy for the specific system of interest. That is, if we require 30

31 [ Γ ] + [ Γ ] + [ Γ ] [ ] T Vne Vee Eg. s. v; For the system of interest, then we clearly can t get too low an energy. This gives the variational principle: E = min T Γ + V Γ + V Γ [ ] [ ] [ ] gs.. ne ee { Γ E[ Γ ] E ˆ g. s. for the H of interest} For an arbitrarily Hamiltonian, though, E T Γ + V Γ + V Γ min [ ] [ ] [ ] gs.. ne ee { Γ E[ Γ ] E ˆ g. s. for one arbitrary H} Since Egs = min T Γ + Vne Γ + Vee Γ { Γ E[ Γ ] E ˆ g. s. for the H of interest} but, in general, Egs.. min T Γ + Vne Γ + Vee Γ { Γ E[ Γ ] E ˆ g. s. for one arbitrary H} the ground-state energy is obtained by [ ] [ ] [ ].. [ ] [ ] [ ] 31

32 Egs = max min T Γ + Vne Γ + Vee Γ [ ] [ ] [ ].. Hˆ { Γ E[ Γ ] E ˆ g. s. H } The maximizing Ĥ is the energy operator for the system. This is not a practical procedure one has to solve the many-fermion problem many times to do the outer maximization. It is better to just solve it once, outright. Proof by picture 3

33 33

34 One-Electron Density Matrix * ( ; ) = Ψ (,,, ) Ψ(, ) γ r r r r r r r r dr r For a Hamiltonian of the form ( r ) ( ) Hˆ = tˆ + v r + i i i= 1 i= 1 j i ri rj model with Hamiltonian ĥ. 34 The energy can be written in terms of the first-order density matrix as: E [ ] ( ) ˆ tv,, γ = δ r1 r 1 t( r1) + v( r1) γ ( r1; r 1) dr1dr 1+ Vee [ γ] E [ γ] = Tr hˆ γ + V [ ] h ˆ, ee γ The variational principle is: E ˆ ˆ gs.. h, min Tr hγ = + V Define E min hˆ 1 ee[ γ ] γ from Ψ as the ground-state energy of the independent particle

35 E min hˆ = min antisymmetric Ψ Ψ ˆ h ri i= 1 ( ) ΨΨ Ψ Theorem (Garrod and Percus): γ (, ) and only if for every ĥ. r r is (ensemble) -representable if 1 1 Tr hˆγ E hˆ min E ˆ gs.. h, min Tr h = + V ˆ γ ee[ γ ] ˆ ˆ ˆ { γ Tr hγ E min h for every h} 35

36 E ˆ gs.. h, min Tr h = + V ˆ γ ee[ γ ] ˆ ˆ ˆ { γ Tr hγ E min h for every h} If we choose just one ĥ, then E ˆ gs.. h, min Tr hˆγ + V ee[ γ ] ˆ ˆ ˆ { γ Tr hγ E min h for one specific h} Assertion: The exact ground-state energy is obtained by E ˆ ˆ gs.. h, max min Tr hγ = + V ee[ γ ] hˆ { γ Tr hˆγ E ˆ min h } The maximizing ĥ is an interesting choice for the one-electron Hamiltonian in the mean-field model, because it represents a one-electron energy operator for the system. One-Electron Density ρ ( x) δ( r x ) = Ψ Ψ i= 1 i 36

37 For a Hamiltonian of the form ˆ 1 i H = v( ri ) + + i= 1 i= 1 j i r i r j The energy can be written in terms of the electron density as: Ev, [ ρ] = ρ( x) v( x) dx + F[ ρ] The variational principle is: Egs.. v, = min ρ x v x dx + F Define v [ ] [ ] ( ) ( ) [ ρ ] ρ from Ψ min v as the ground-state energy of the classical structure problem with energy E ( x,, x ) = v( x ) 1 i i= 1 vmin = min v( xi ) = min v( x) x i= 1 x i 37

38 Theorem: ρ ( x ) is (ensemble) -representable if and only if ρ ( x) v( x) dx vmin [ v] for every v( x ). This constraint merely implies that ρ ( x ) 0. If ρ ( ) < 0 then one can obtain a contradiction by letting v( ) while it stays the same elsewhere. x at some point, x at that point, The exact ground-state energy is obtained from Egs..[ v, ] = min ρ x v x dx + F { ρ ρ( x) v ( x) vmin [ v ] for every v } Assertion: The exact ground-state energy is obtained by Egs..[ v, ] = max min ρ x v x dx + F v { ρ ρ( x) v ( x) dx vmin [ v ] } ( ) ( ) [ ρ ] ( ) ( ) [ ρ ] 38

39 Maximizing v( x ) ensures that the energy of the sytem does not get too low. Except for a constant shift, the maximizing v( x ) is a representation of the local energy, E ( x), of the system since requiring ρ ( x) E ( x) dx Egs..[ v; ] is sufficient to enforce the variational principle. 39

40 The Pair Density, Revisited ( x, x ) ( ri x ) ( rj x ) ρ Ψ δ δ Ψ 1 1 i= 1 j i For a Hamiltonian of the form 1 ˆ i H = v( ri ) + + i= 1 j i i 1 i j = r r The energy can be written in terms of the pair density as: 1 v( x1) + v( x) 1 Ev, [ ρ] = ρ( 1, ) d 1d T ρ x x + x x + 1 x1 x The variational principle is: E v, = min E ρ [ ] [ ] gs.. v, ρ from Ψ [ ] ( ) Define V min V as the ground-state energy of the classical structure problem with -body interaction potentials 40

41 V ( ) min x i= 1 j i ( ) ( j) min V xi, x i Assertion: The exact ground-state energy is obtained by Egs[ v, ] = max min E [ ρ ].. v, ( ) V ( ( ) ) ( ) ( ) { ρ ρ 1, V ( 1, ) V min V x x x x } ( ) ( ) Maximizing V x1, x ensures that the energy of the sytem does not get too low. Except for a constant shift, the ( maximimizing V ) ( x1, x ) is a representation of the pairwise interaction energy, E ( x1, x). Sketch of Proof T ρ can be chosen to be convex. Step 1. [ ] Proof: T ρ can be constructed using the Legendre-transform formalism, [ ] 41

42 [ ρ ] ( ) ( ) T = sup E V ; ρ x, x V x, x dxdx ( V ) ( x1, x) This formalism always gives convex functionals. ( ) ( ) Step. The energy functional can be chosen to be convex. Proof: 1 v( x1) + v( x) 1 Ev, [ ρ] = ρ( 1, ) d 1d T[ ρ] x x + x x + 1 x1 x E ρ is the sum of a linear functional and a convex functional, it Since [ ] is convex. v, 4

43 Step 3. The set of all (, ) convex set. ρ x x that give too small an energy is an open, L 1 { ρ E [ ρ ] E } < v, v, gs.. Proof: The energy is a convex functional. For any convex functional, the set of arguments for which the function is less than or equal to some value is convex. ρ x, x is closed and convex. Step 4. The set of all -representable ( ) 1 { ρ ρ is ensemble -representable} Proof: The proof is a standard exercise, and follows from the fact any ensemble--representable pair density can be written as ρ δ δ * [ Γ; x, x ] Tr ( x ri) ( x rj) wiψi( z z) Ψi ( z i= 1 j i i i w i = 1; 0 w 1 i 43

44 Step 5. The sets of -representable pair densities and pair densities with too low an energy do not intersect. Proof: E ρ = E min v, gs.. ρ [ ] and so no -representable ρ is in E < E L v, { ρ v, [ ρ ] gs..} ( ) ( 1, ) ( ) L ( ) ρ ( ) Step 6. There must exists some V x x such that ρ ( ) ( ) ( ) x1, x V x1, x dx1dx x1, x V x1, x dx1dx For all ρ and ρ L in L v,. Proof: This is the so-called geometric Hahn-Banach Theorem. Step 7. The constant of separation in this result is ( ) V min, the solution to the classical structure problem. Proof: A bit complicated; it is the Banach-space analogue of the polar cone theorem used by Garrod and Percus. 44

45 Acknowledgements Discussions: Prof. Ernest Davidson Prof. Mel Levy University of orth Carolina (Chapel Hill) University of Washington (Seattle) orth Carolina A&T University Funding: SERC, CFI/OIT, PREA, CRC, McMaster U As an adolescent I aspired to lasting fame, I craved factual certainty, and I thirsted for a meaningful vision of human life so I became a scientist. This is like becoming an archbishop so you can meet girls. Matt Cartmill 45