Does Møller Plesset perturbation theory converge? A look at two electron systems

Transcription

1 Does Møller Plesset perturbation theory converge? A look at two electron systems Mark S. Herman George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry Virginia Polytechnic Institute and State University Blacksburg, Virginia , U.S.A. March 8, 008 Abstract We study convergence or divergence of the Møller Plesset Perturbation series for systems with two electrons and a single nucleus of charge Z > 0. This question is essentially to determine if the radius of convergence of a power series in the complex perturbation parameter λ is greater than. The power series is centered at λ = 0, so we try to find whether or not the singularity closest to λ = 0 is inside the closed unit disk in the complex plane. We give a description of possible causes for divergence in the general problem and then examine two Helium like models. The first model is a simple one-dimensional model with delta functions in place of Coulomb potentials. The second is the realistic three-dimensional model. For each model, we show rigorously that if the nuclear charge Z is sufficiently large, there are no singularities for real values of λ between and. Using a finite difference scheme, we present numerical results for the delta function model. The numerics are consistent with proven results and also suggest that the closest singularity occurs where λ is real and negative. Partially Supported by National Science Foundation Grant DMS

2 Introduction Møller Plesset perturbation theory MP theory) [] is a commonly used technique to obtain corrections to the Hartree-Fock energy for an atom or molecule. The Hartree- Fock ground state is an eigenfunction of a specific Hamiltonian H 0 that is the sum of -electron Fock operators. H 0 is not the physical Hamiltonian of interest H phys. If we define V = H phys H 0, then MP theory is the application of perturbation theory to the operator Hλ) = H 0 + λv, to approximate the eigenvalue E) of H) = H phys. The question of convergence or divergence is to determine whether or not the radii of convergence of the power series expansions of the ground state eigenvalue Eλ) = E k λ k and ground state eigenfunction Ψλ) = Ψ k λ k of Hλ) are greater than k=0 one. Although MP theory was developed many years ago, due to computational barriers, the question of possible divergence of the MP series has only recently been studied. Advances in technology have allowed for the computation of high-order terms in the MP series. Using a variety of basis sets, Knowles, Handy, et al. [, 3], computed MP energies up to as much as 48 th order for a variety of molecules. They concluded that the series were likely converging, but slowly or erratically in some cases. More recently, several instances of divergent behavior have been observed [4, 5, 6, 7, ]. Olsen and collaborators [4, 5, 6] discovered divergent behavior in MP series for several molecules and basis sets. They concluded that the divergence was the result of a back door intruder, that is, a crossing involving the eigenvalue Eλ) with a non-physical excited state at some λ 0 with Reλ 0 ) < 0. They found the choice of basis set to be of importance in the emergence of the intruder state. Stillinger examined this question for two electron atoms of nuclear charge Z > 0 [8]. He proposed that the eigenvalue Eλ) is an analytic function of λ in a neighborhood of the origin, but that there exists a critical point λ c located on the negative real axis. The critical point λ c is a singularity of Eλ), and if λ λ c, then Eλ) no longer exists as a bound state eigenvalue of Hλ). Furthermore, λ c corresponds to a threshold at which the molecule can be thought to dissociate into an electron electron bound state running away from the nucleus. Clearly, if λ c >, this phenomenon would cause divergence of the MP series. The electron electron bound state is the bottom of the continuous spectrum of Hλ), and the critical point λ c is a threshold at which k=0

3 Eλ) is absorbed into the continuous spectrum. In Section we provide more detail regarding the structure of the spectrum of Hλ). We note that contrary to [9], we see no obvious way to prove rigorously that such a critical point always exists in general. It is conceivable that the structure of ν HF to be defined in Section ) would allow an eigenvalue to stay below this threshold for all λ < 0. Sergeev, Goodson, et al. [9, 0,, ] have further analyzed this problem within the context of finite basis sets. They explained the connection between the back door intruder state found by Olsen and the critical point predicted by Stillinger. Once a finite basis set is chosen, one is no longer working with Hλ), but rather an approximate Hamiltonian H ap λ). The operator H ap λ) is a matrix and has no continuous spectrum. Continuous spectrum of Hλ) will be represented by closely clustered eigenvalues of H ap λ). So the critical point λ c at which Eλ) hits the continuous spectrum of Hλ) is represented by a crossing between E ap λ) the eigenvalue of H ap λ) that approximates Eλ) ) and another eigenvalue of H ap λ). So it is possible that the back door intruder seen by Olsen actually corresponds to a continuum state of Hλ), but is represented by an eigenvalue of H ap λ). An analysis of several molecules [] concluded that basis sets that did not include diffuse functions could not model this dissociation phenomenon and as a result no critical point was found in this case. Even standard basis sets that included diffuse functions could not model the electron electron bound state. However, a critical point corresponding to complete dissociation of all valence electrons was found when diffuse functions were included. Upon analyzing the Ar atom, a critical point corresponding to ionization of one electron appeared to be present at some λ c with Reλ c ) > 0 a front door intruder ). It is of interest to study the divergence question for the exact Hamiltonian Hλ). We presume that the better a basis set approximates the space, the more the spectrum of H ap λ) will resemble that of the exact Hamiltonian Hλ). We believe a rigorous mathematical description of this problem would be useful in understanding the scope of the divergence question. We first give a mathematical description of all possibilities that could cause divergence in the MP series in Section and then concentrate on two electron systems with a single nucleus of charge Z > 0. We present numerical results for a simplified -dimensional model with delta functions in place of Coulomb potentials and then move to the physical model. In both models we rigorously prove 3

4 lower bounds on Z for which the Stillinger critical point does not cause divergence. In particular, for a two electron atom with nuclear charge Z, the Stillinger critical point does not cause divergence if Z >.85. Since we first became interested in this topic, we have learned of various proposals to modify the perturbation theory to obtain convergence in certain situations. See, e.g, [3, 4, 5, 6] and the papers referenced therein. We note that successfulness of these proposals has been only been assessed by numerical computations for specific systems in finite basis sets. Possibilities for divergence Consider the Hamiltonian of a molecule of N electrons with positions x i moving in a field of M nuclei with fixed positions R j and charges Z j > 0: H phys = N i= xi N i= M j= Z j x i R j + N i= N j>i x i x j We assume we know the Hartree-Fock wave function Ψ 0, a Slater determinant constructed from known one electron functions {u i } N i=. We then construct the Hartree- Fock potential ν HF and Fock operator F : [ N νi HF uj y) ) φx i ) = dy x i y j= φx i ) ) u j y) φy) dy x i y u j x i ) ], F i φx i ) = x i φx i ) M j= Z j x i R j φx i) + ν HF i φx i )..) Define H 0 = N i= F i and V = H phys H 0, and consider Hλ) = H 0 + λ V : Hλ) = N i= xi N i= M j= Z j x i R j + N i= ν HF i + λ N i= N j>i x i x j N i= ν HF i.) If Ψ 0 is the Hartree-Fock wave function, and ε i are the Lagrange multipliers from 4

5 the Hartree-Fock equations, then H 0 Ψ 0 = E 0 Ψ 0, where E 0 = N i= ε i. Here E 0 and Ψ 0 are the ground state eigenvalue and eigenfunction of H 0. As described earlier, MP theory is the use of perturbation theory to find the power series expansions Ψλ) = Ψ k λ k and Eλ) = E k λ k corresponding to the ground state of Hλ), k= k= satisfying Hλ) Ψλ) = Eλ) Ψλ). Since H) = H phys, one uses the power series to find E) and Ψ). Of course, this is possible only if the radii of convergence of the series are greater than, or equivalently if Eλ) is an analytic function of λ C for λ [7]. Under specific hypotheses, analyticity of Eλ) is guaranteed as long as Eλ) remains an isolated point of the spectrum of Hλ). For details see Section XII. of [8]. If we could show that dist{ Eλ), σ Hλ) ) Eλ) } > 0 for every λ C such that λ, then the power series of Eλ) would have a radius of convergence larger than. Divergence can occur if either Eλ) runs into the continuous spectrum of Hλ) at some λ, or if Eλ) is involved in a level crossing at some λ off the real axis. For details regarding the spectrum of operators, see for example [9]. We must determine the structure of the spectrum of Hλ) for λ. If λ R, Hλ) is self-adjoint, and the well-known HVZ theorem gives us a description of the continuous spectrum. See Section XIII.5 of [8]. The continuous spectrum is determined from the spectrum of Hamiltonians of different cluster decompositions. For molecular systems with fixed nuclei, the bottom of the continuous spectrum given by the HVZ theorem corresponds to an ionization threshold. We present two examples for maximal clarity. Although they deal with specific systems, they are intended to indicate clearly how the process generalizes. We note that the usual formulation of the HVZ theorem does not actually apply here since the Hamiltonians of interest involve ν HF, which is not a multiplication operator in the presence of exchange terms. In Example we assume that the HVZ theorem can be generalized to handle this case, although it is not obvious that it can be. This is not an issue in the proofs of Theorems 3. and 3. below) since they do not involve exchange terms. 5

6 i) Example Consider a Lithium like atom with a nucleus of charge Z > 0 located at the origin. Then.) becomes Hλ) = 3 i= xi Z 3 i= x i + 3 i= ν HF i + λ x x + x x 3 + x x 3 To find the continuous spectrum of Hλ), we look at the following Hamiltonians of cluster decompositions: D = {Z}{3}: 3 i= ν HF i The system with x 3 located a large distance from the origin, while x and x are near the origin. We consider H D λ) = D = {Z}{3}: i= xi Z i= x i + i= ν HF i + λ x x i= ) ν HF i The system with x and x 3 located a large distance from the origin, but near each other, while x is located near the origin. We consider H D λ) = x Z ) + ν HF + λ ν HF ) + x x + λ x where x = x x 3, and we have removed the center of mass of the {, 3} cluster. D 3 = {Z}{}{3}: The system with x and x 3 located a large distance from the origin and a large distance from each other, while x is located near the origin. We consider D 4 = {Z}{}{3}: H D3 λ) = x Z x + ν HF + λ ν HF ). The system with x and x 3 located a large distance from the origin, but near each other, while x is located a large distance from the origin, x, and x 3. We consider H D4 λ) = x + λ x, where x = x x 3, and again we have removed the center of mass of the {, 3} cluster. D 5 = {Z}{3}: The system with x, x, and x 3 located a large distance from the origin, but near each other. We consider H D5 λ) = x 3 4 x + λ x 6 + x x + ) x +, x. ). ),

7 where x = x x 3, x = x x + x 3, and we have removed the center of mass of the {,, 3} cluster. D 6 = {Z}{}{}{3}: The system with x, x, and x 3 located a large distance from the origin and a large distance apart. This cluster decomposition contributes the halfline [0, ) to the continuous spectrum of Hλ). For fixed λ R, the HVZ theorem says that if E Dk is an eigenvalue of H Dk λ), then the half-line { E Dk + t : t 0 } is in the continuous spectrum of Hλ). The continuous spectrum of Hλ) is σ c Hλ)) = [0, ) 5 k= { E Dk + t : E Dk is an eigenvalue of H Dk λ), t 0 }..3) Since Hλ) and the H Dk the half-lines { E Dk are self-adjoint when λ R, they have real eigenvalues. So + t : t 0 } are overlapping, and.3) can be written σ c Hλ)) = [Σλ), ), where Σλ) = min k [ inf σ H Dk λ) ) ]..4) We note that if λ 0, the D, D 4, and D 5 cluster decompositions will not determine σ c Hλ) ). But if λ < 0, H D λ), H D4 λ) and H D5 λ) very well could have eigenvalues below their continuous spectra, so these decompositions can determine σ c Hλ) ) in this case. It is apparent that clusters that involve only electron coordinates must also be considered in determining the continuous spectrum of Hλ). This is the origin of the critical point predicted by Stillinger [8]. This is a consequence of the HVZ theorem and purely a mathematical issue. Although Hλ) is clearly not self-adjoint if λ has non-zero imaginary part, the HVZ theorem is generalized to the case where Imλ) 0 by Proposition of Section XIII.0 of [8]. In this case the continuous spectrum of Hλ) still takes the form of.3) σ c Hλ) ) = [0, ) 5 k= We provide a second example illustrating this. { E Dk + t : E Dk is an eigenvalue of H Dk λ), t 0 }..5) ii) Example Let x, y R 3 and H 0 = x + y ) x + ) y and V = x + ), y 7

8 and consider the Hamiltonian Hλ) = H 0 + λv = x + y ) + λ) x + ). y From the well-known eigenfunctions of the Hydrogen atom Hamiltonian, we can find the eigenfunctions of x + λ) x in terms of λ. The eigenfunctions are in L R 3 ) if and only if Reλ) >. We see that if Reλ) >, x + λ) { } x has eigenvalues +λ) and continuous spectrum [0, ). So, if Reλ) > n n=, Imλ) = 0, Hλ) has eigenvalues { +λ) + n ) } n= that are isolated from the continuous spectrum [ +λ), ). Figure shows the spectrum of Hλ) in the complex plane for λ = 0. If Reλ) >, Imλ) 0, then Hλ) has eigenvalues { +λ) m + n ) } n,m= that are isolated from the continuous spectrum. The continuous spectrum is made up { } of [0, ) and half-lines of the form +λ) + t : t 0, for n N. Figure n shows the spectrum of Hλ) in the complex plane for λ = 0 i. If Reλ), no eigenvalues exist and Hλ) has only continuous spectrum of [0, ). We note that the eigenvalues approach the continuous spectrum as Reλ) approaches. We also note that the power series centered at λ = 0 representing the eigenvalues, are power polynomials and have a radius of convergence of. The eigenvalues approach the continuous spectrum near Reλ) = and no longer exist past Reλ) =, but this does not cause a singularity in the functions that represent the eigenvalues when they exist. We expect that a collision between the eigenvalue and the continuous spectrum usually does cause a singularity in the eigenvalue, but showing such a collision occurs is not sufficient to prove that a singularity exits. This example should be illustrative of the complexity of this problem. When Imλ) 0, each eigenvalue of a given cluster Hamiltonian contributes a half-line to the continuous spectrum of Hλ). In principle this could allow for infinitely many halflines to be considered as possible culprits causing divergence in the MP series. We already mentioned that in practice the divergence issue has been found to be basis set dependent. In the absence of any general theorem regarding divergence/convergence of the MP series, this implies that divergence must be considered on a case-by-case basis. Even in small molecules this would likely make it impractical to consider continuum thresholds of each cluster decomposition individually. There has been much work devoted to classifying and approximating the location of the singularities of Eλ) 8

9 based on the terms of the MP series [9, 0, ]. This appears much more practical. 3 Rigorous results for real λ i) The delta function model We now concentrate on Helium like systems with a nucleus of charge Z > 0 located at the origin. We first look at a simplified dimensional model with delta functions in place of Coulomb potentials. This model will provide insight into the physical model we analyze later. Aside from the fact that many of the quantities of interest are exactly solvable in the delta function model, numerics are easily obtainable since it is -dimensional. The Hamiltonian of the system is H D = ) x + x Z ) δx ) + δx ) + δx x ), 3.) where x and x are the electron space coordinates -dimensional) and Z is the nuclear charge. The restricted Hartree-Fock wave function Ψ d 0) can be written as a product, Ψ d 0)x, x ) = φ 0 x ) φ 0 x ), 3.) and anti-symmetrized by way of spin functions. The Hartree-Fock equation is d dx φ Z δx) φ + φ3 = γ φ. 3.3) For Z > /, 3.3) can be solved exactly with unique normalized solution [0] φ 0 x) = Z 4 Z with γ 0 = Z. ) In this case.) becomes νi HF = exp Z ) ) x 4 Z exp Z φ 0 y) δx i y) dy = φ 0x i ), ) x ), 3.4) F i = d dx i Z δx i ) + φ 0x i ). 3.5) From 3.3) we see that F i φ 0 x i ) = γ 0 φ 0 x i ). Analogous to.), we define 9

10 H d λ) = x ) + x Z [ δx ) + δx ) ] + φ 0x ) + φ 0x ) + λ [ δx x ) φ 0x ) + φ 0x ) ) ]. 3.6) We have H d ) = H D, and from 3.) and 3.3), H d 0) Ψ d 0) = γ 0 Ψ d 0). I.e., E d 0) = γ 0 = Z ) and Ψ d 0) = φ 0 x ) φ 0 x ) are the zeroth order ground state energy and wave function, respectively. Let E d λ) and Ψ d λ) be the eigenvalue and eigenfunction of H d λ) that correspond to the ground state when λ R. As in example, we must examine Hamiltonians of different cluster decompositions to gain information about σ c H d λ) ). There are only two cluster decompositions that are of concern. Define H SI λ) = d dx Z δx) + φ 0x) + λ φ 0x) ), 3.7) the singly ionized Hamiltonian representing a bound state between the nucleus and one electron. Let x = x x, and define H ee λ) = x + λ δx ), 3.8) the Hamiltonian after center of mass removal), representing a bound state between the two electrons far from the nucleus. We refer to this as the electron electron bound state Hamiltonian. We see that H ee λ) is just the Hamiltonian of a delta well potential of strength λ, so if Reλ) < 0, it has one eigenvalue E ee λ) = λ ) No eigenvalues exist if Reλ) 0 note that 3.9) is true for complex λ, not just real λ). The value λ c on the negative real axis for which lim λ λ c E d λ) E ee λ) = 0, is the critical point predicted by Stillinger [8]. For now, we assume that λ is real, unless stated otherwise. theorem as in.4), we know the continuous spectrum is Applying the HVZ σ c H d λ) ) = [ Σλ), ), where Σλ) = min { 0, inf σ H SI λ) ), inf σ H ee λ) ) }. 3.0) Applying the HVZ Theorem to H SI λ) and H ee λ) we get σ c H SI λ) ) = σ c H ee λ) ) = [ 0, ). 0

11 Let E SI λ) be the lowest eigenvalue of H SI λ), if any exist. Then inf σ H SI λ) ) = min { 0, E SI λ) } See sketch in Figure 3. inf σ H ee λ) ) = min { 0, E ee λ) } Theorem 3.. If Z > , 3.) then E d λ) < Σλ) for λ. In particular, E d λ) remains an isolated point of the spectrum of H d λ) for λ. The details of the proof are presented in the Appendix. We provide an outline here. The proof relies heavily on the fact that E d λ) 0 inside the region of analyticity, which comes from perturbation formulas. It follows that the graph of E d λ) lies below its tangent lines. The tangent line to E d λ) at λ = 0 can be found exactly. We refer to this tangent line as Lλ). The proof is separated into the two cases that can cause divergence, the singly ionized threshold E SI λ) and the electron-electron bound state threshold E ee λ):. We show that Lλ) < E SI λ) for 0 λ, provided Z > + 3 ) We use E SI λ) 0 also coming from perturbation formulas), which implies that the graph of E SI λ) lies above its secant lines. We show that the secant line passing through 0, E SI 0) ) and, E SI ) ) is above Lλ). See sketch on Figure 4. We then use a simple argument to show E d λ) < E SI λ) for λ < 0.. We show that Lλ) < E ee λ) for λ < 0, provided Z is above the bound in 3.). This is simple since we are able to obtain Lλ) and E ee λ) exactly. See sketch on Figure 5. From this, it easily follows that E d λ) < 0, ruling out the threshold at 0 as a possible issue. Figure 6 shows a qualitative sketch of the eigenvalue and thresholds in question, along with the tangent and secant lines that are relevant to the proof.

12 ii) The physical model We now turn our attention to the physical model of Helium like atoms. The Hamiltonian of the system is H phys = x + x ) Z x + ) x + x x, 3.) where again x and x are the electron space coordinates 3-dimensional) and Z > 0 is the nuclear charge. As before, the restricted Hartree-Fock wave function Ψ ph 0) can be written as a product, Ψ ph 0)x, x ) = φ 0 x ) φ 0 x ), 3.3) and anti symmetrized by way of spin functions. The Hartree-Fock equation is φx) Z ) φy) x φx) + x y dy φx) = γ φx). 3.4) While the solution to 3.4) is not known explicitly, as in the delta model, it has been proved [] that for Z >, there exists φ 0 with φ 0 =, such that Ψ ph 0) from 3.3) minimizes Ψ, H phys Ψ over all such restricted Slater Determinants. In addition, φ 0 must solve 3.4) with eigenvalue γ 0 < 0. Define the Hartree-Fock potential ν HF and the Fock operator F to be ν HF i = φ0 y) x i y dy, F i = x i Z x i + ν HF i, 3.5) We have F i φ 0 x i ) = γ 0 φ 0 x i ), and it has been proved [] that γ 0 = inf{σf i )}. Again consider H ph λ) = H 0 + λv where H 0 = F + F, V = x x ν HF + ν HF ). 3.6) Analogous to the delta model, we have H ph ) = H 0 + V = H phys, with E ph 0) = γ 0, and Ψ ph 0) the zeroth order ground state energy and wave function, respectively. Let E ph λ) and Ψ ph λ) be the eigenvalue and eigenfunction of H ph λ) that correspond to the ground state when λ R. As before, we must show dist { E ph λ), σ c H ph λ) ) } > 0 if λ R. Define the singly ionized and electron electron bound state Hamiltonians as in 3.7) and 3.8): H SI λ) = x i Z x i + ν HF i + λ ν HF i ), 3.7)

13 H ee λ) = x + λ x, 3.8) where x = x x. Here H ee λ) is a hydrogenic Hamiltonian with charge Z = λ. The lowest eigenvalue is E ee λ) = λ 4 if Reλ) < 0, and no eigenvalues exist for Reλ) 0. Also, in this formulation 3.0) still holds. Note that if λ is complex, we need to consider all of the bound states of H ee λ), namely E n) ee λ) = λ 4n. We discuss this later. Theorem 3.. If Z > ) 4.85, 3.9) then E ph λ) < Σλ) for λ. In particular, E ph λ) remains an isolated point of the spectrum of H ph λ) for λ. The details of the proof are presented in the Appendix. The proof is similar to the proof of Theorem 3. in the delta function model. Many of the corresponding quantities that were exactly solvable in the delta function model are no longer explicitly obtainable, but we are able rigorously to obtain 3.9) with the use of trial functions. It is apparent that the eigenvalue and thresholds behave qualitatively the same as in the delta function model, illustrated in Figure 6. 4 Numerical results for the delta function model for real λ We have numerical results for the delta model from an elementary finite difference scheme. The space is discretized into grid points, so the continuous variables x, x have become discrete variables. The derivative operators are approximated by matrices via difference quotients. The problem is constrained to a box, with boundary conditions that require the eigenfunctions go to zero on the boundary. The Hamiltonians are all matrices, so all spectrum comes in the form of eigenvalues. Let h d λ) and h SI λ) be the matrix approximations to H d λ) and H SI λ), respectively. Clusters of eigenvalues of h d λ) correspond to continuous spectrum σ c H d λ)) of the actual operator. By looking at a contour plot of an eigenvector corresponding to a specific low-lying eigenvalue of h d λ), we can determine if it corresponds to a bound state eigenvalue or if it corresponds 3

14 to a piece of continuous spectrum of H d λ). Negative energy states corresponding to the continuous spectrum are singly ionized states, or electron electron bound states. All plots shown are for Z =.38, corresponding to the value in 3.). For a variety of choices of λ, we computed several of the eigenvalues of h d λ) with smallest real parts. Figures 8, 9, and 0 show contour plots of the absolute squares of some eigenfunctions of h d λ) at different values of λ. The functions are approximately zero outside the regions where the contours lie. In Figure 7, the lowest curve on approximately. λ corresponds to E d λ). To the left of about λ =. it is absorbed into the continuous spectrum of H d λ). Figure 8 shows a plot of the eigenfunction squared of this lowest eigenvalue at λ = 0.5. We can see it is a bound state with the probability density near zero outside a region around the origin. This is a state with both electrons near the nucleus. The lowest curve on Figure 7 from about λ. corresponds to E ee λ). Figure 9 shows the eigenfunction squared of the lowest state at λ =.85. Notice that the probability density is near zero outside regions along x = x with x and x far from zero. This is a state where both electrons are near each other but away from the nucleus fixed at the origin. The second lowest eigenvalue on about 0.3 λ corresponds to the E SI λ) threshold. This is a singly ionized state. Figure 0 shows a plot of the eigenfunction squared at λ = 0. The probability density is near zero outside regions along the coordinate axes and away from the origin. This is a state where one electron is near the nucleus, while the other is far away. 5 Considering complex λ As mentioned earlier, we need to show E d λ) is an isolated point of σh d λ)) for all λ, so we must consider complex λ. If λ is not real, in addition to showing that E d λ) stays away from σ c H d λ)), we must also consider the possibility of level crossings upsetting analyticity of E d λ). As we already mentioned, the usual HVZ theorem is not applicable for non-self-adjoint operators but it is generalizable to handle H d λ) and H ph λ) in the case Imλ) 0, taking the form of.5). See the sketch in Figure. i) Numerical results for the delta function model with λ = α + iβ, β 0 The structure of σ c H d λ)) for λ = i can be seen in the numerical approxi- 4

15 mation in Figure. Notice the cluster of eigenvalues starting near E ee λ) and running off to infinity towards the right. If we plotted the eigenvectors corresponding to these eigenvalues, we would see that these are electron electron bound states. The cluster starting near the point i corresponds to the half-line from the E SI λ) threshold. If we plotted the eigenvectors, we would see these are singly ionized states. The eigenvalue located near i appears to be the one and only bound state of h d λ), corresponding to E d λ). Again we would see this if we plotted the eigenvector. If there is only one bound state in the delta model at each λ {λ : λ }, then level crossings are not a concern. One would only need to show that E d λ) remains a positive distance from σ c H d λ)) at each λ. We note that this is not true in the physical model since Helium is known to have infinitely many bound states. So in the physical model, one would need to rule out the possibility of level crossings causing a singularity in E ph λ). Contour plots in the complex λ plane, of the real and imaginary parts of the eigenvalue of h d λ) with smallest real part, are shown in Figures 3 and 4. Again we chose Z =.38. The plot is given in the upper half-plane only since we know E d λ) = E d λ) by the Schwartz Reflection Principle [7]. Only the sign of the imaginary part of E d λ) changes in the lower half-plane, i.e., points upsetting analyticity come in conjugate pairs. The unit circle is included in the plot since it is the region of interest. Since we know that E d λ) has smallest real part on the real axis compared to the rest of σh d λ)) at least for large enough λ and Z as shown earlier), the smoothness of the plots suggest that this is a plot of E d λ). Furthermore it appears that E d λ) is analytic and it remains away from σ c H d λ)) as long as Reλ) >. The noise near Reλ) = suggests either a collision of E d λ) with the half-line from the E ee λ) threshold and so also a collision with σ c H d λ) ) ) or a different eigenvalue of h d λ) is becoming the one with smallest real part. Recall from Figure 7, E d λ) appears to collide with σ c Hλ) ) near λ =.. Let α = Reλ), β = Imλ). Notice from Figure 4 that Im E d λ)) is negative and decreasing as β increases. Note that Im E d λ) ) = β Ψ dλ), V Ψ d λ) Ψ d λ), Ψ d λ) implies 5

16 sgn { Im E d λ) ) } = sgn { β Ψ d λ), V Ψ d λ) } = sgn { β [ Ψ d λ), δx x ) ν HF ) Ψd λ) Ψ d λ), ν HF Ψ d λ) ] }. We believe that Ψ d λ), δx x ) ν HF ) Ψ d λ), which is 0 at λ = 0, remains small, and thus sgn { Im E d λ) ) } = sgn { β Ψ d λ), ν HF Ψ d λ) } = sgn {β}. Since sgn {Im E ee λ))} = sgn { αβ} = sgn {β} for α < 0, it appears that Im E d λ)) and Im E ee λ)) have opposite sign. So if β 0, E d λ) cannot approach the piece of σ c H d λ)) coming from the E ee λ) threshold. The plots of Re E SI λ) ) and Im E SI λ) ) given in Figures 5 and 6 show that E SI λ) exhibits behavior similar to that of E d λ). Notice that the contours of Re E d λ)) and Re E SI λ)) are nearly horizontal. The real parts change little as β increases. Since Re E d λ)) is lower than Re E SI λ)) for λ = α and appropriate nuclear charge Z > 0, we conclude that it remains as such for λ = α + iβ inside the unit disk. ii) Relating the Delta Model to the Physical Model Since the structures of the delta and physical models are rather similar, it is natural assume that E d λ) in the physical model behaves much the same in relation to the pieces of σ c H ph λ)) which arise from the bound states of H SI λ) and H ee λ). Although there are infinitely many bound states of H ee λ) when Reλ) < 0 in the physical model, we know them explicitly from 3.8) to be E n) ee λ) = λ 4n. So, sgn { Im E n) ee λ) ) } = sgn {β}, and we expect that Im E ph λ) ) and Im E ee n) λ) ) probably have opposite sign inside {λ : λ } if Z > ) Furthermore, Re Eph λ)) and Re E SI λ)) probably change little as β increases as in the delta model. We also believe that the second derivatives of E ph λ) and E SI λ) are small as suggested by Figure 7). If one could show these quantities were approximately linear, it could lead to a proof of the necessary result. The only remaining issue would then be to remove the possibility of level crossings involving E ph λ) with other bound states of H ph λ) at some complex λ in the unit disk. 6

17 Since only one bound state appears in the delta model, we cannot use it to obtain any insight into this problem. Do the other bound states act similar to E ph λ) when λ is complex? If so, we would conclude there is no crossing involving E ph λ) and most likely it is analytic inside the unit disk. The underlying complexity of this type of analysis is mainly due to the non-selfadjointness of the Hamiltonians when λ is not real. This makes it difficult to prove any useful properties of the bound states or thresholds. Any upper bounds on Re E ph λ) ) or Im E ph λ) ) or their derivatives for that matter) seem not easily obtainable. It should be noted that even if there exists λ 0 with λ 0 at which there is a level crossing involving E ph λ), or where E ph λ) is absorbed into H ph σ c λ)), it does not necessarily cause a singularity in the function E ph λ). It may be that E ph λ 0 ) is not an eigenvalue of H ph λ 0 ) but E ph λ) is analytic at λ = λ 0. Example has a simple illustration of this possibility. The behavior of eigenvalues with regard to analyticity) near points at which they absorb into continuous spectrum has been previously analyzed under certain hypotheses in [] and [3]. 6 Conclusion We examined the question of Møller Plesset convergence for Helium like systems. Singularities in the eigenvalue cannot occur if the eigenvalue remains an isolated point of the spectrum of the perturbed Hamiltonian H ph λ). For λ R, one need only show that the eigenvalue remains a positive distance from the continuous spectrum of H ph λ). The continuous spectrum can be described in terms of bound states of ionized Hamiltonians. As previously conjectured, there may exist a singularity in the eigenvalue at some negative value of the perturbation parameter λ, at which the eigenvalue approaches a threshold corresponding to an electron electron bound state. This is not the only possible occurrence of a singularity however. There may be singularities arising from this electron electron bound state at some complex λ in the left half-plane. The singly ionized state, a bound state between one electron and the nucleus, may also cause divergence. When λ is complex, it is also possible that the existence of level crossings can cause a singularity in the eigenvalue in question. The simplified delta function model considered here gives insight into the nature of the physical problem. In the delta function model we showed that if Z > , 7

18 then there does not exist a singularity at any λ R with λ. Using a similar approach, we showed the same for the physical problem if Z > ) While there is not yet a proof excluding the possibility of singularities at complex values of λ inside the unit disk, the numerics provided in the delta function model are encouraging. The eigenvalue appears to move away from these thresholds as the imaginary part of λ increases. It seems apparent that there is at most one bound state for each of H d λ), H SI λ), and H ee λ) in the delta model. So level crossings are not a concern, and we only need to know the distance from the eigenvalue to two thresholds. This is dramatically simpler when compared with the physical problem where each of these Hamiltonians has infinitely many bound states. Level crossings are certainly an issue in this case and there are infinitely many pieces of σ c H ph λ) ) with thresholds associated with each of the bound states of H SI λ) and H ee λ). While this gives much insight into the nature of the convergence of Møller Plesset for the Helium atom, this is a far cry from obtaining any sort of general result concerning atoms or molecules. Even the Lithium atom problem is much more complicated for various reasons, not the least of which is spin considerations. Appendix Here we present the details of the proofs of Theorem and Theorem. Proof of Theorem 3.: We first need some perturbation formulas. Let H d λ) = H 0 + λv, where H 0 and V are defined by 3.6). Inside the region of analyticity of E d λ) we have H d λ)ψ d λ) = E d λ)ψ d λ). We can assume Ψ d λ) is normalized. By differentiating both sides of this eigenvalue equation we obtain E d 0) = Ψ d0), V Ψ d 0) and Ψ d 0) = H 0 E d 0) ) r P V Ψ d 0), where H 0 E d 0) ) r A-) is the reduced resolvent, defined on Ψ d 0), the subspace perpendicular to Ψ d 0), and P = I Ψ d 0) Ψ d 0) is the projection onto Ψ d 0). 8

19 By differentiating a second time and using A-), we obtain E d 0) = Ψ d0), V H 0 E d 0) ) r P V Ψ d 0) = P V Ψ d 0), H 0 E d 0) ) r P V Ψ d 0) A-) 0, since H 0 E d 0) ) r is a positive operator on Ψ d 0). This is a general argument that only requires H 0 and V to be self-adjoint and E d λ) = inf σ H d λ) ) > to be a simple discrete eigenvalue. So, if λ R and E d λ) is isolated at the bottom of σh d λ)), then E d λ) 0. A-3) From A-3) it follows that the graph of E d λ) lies below its tangent lines. Let Lλ) be the tangent line to E d λ) at λ = 0. Lλ) goes through the points 0, E d 0) ) and, E d 0) + E d 0) ). Note that E d0) + E d 0) is the usual HF energy. Recall from 3.4) that we have E d 0) = γ 0 = Z ). So, using 3.4), we have E HF =: E d 0) + E d 0) = = Z ) + Ψ d 0), V Ψ d 0) Z ) + φ 0x ) φ 0x ) δx x ) φ 0x ) φ 0x ) ) dx dx = Z ) φ 4 0x) dx = Z ) Z ) 6 = Z + Z. A-4) From this we have Lλ) = Z + ) λ Z ). A-5) 6 9

20 The singly ionized threshold We now show that Lλ) < E SI λ) for λ, provided Z > + 3 ) With 3.5) and 3.7) we see that Ψ SI 0) = φ 0 and E SI 0) = Z ) solve H SI λ) Ψ SI λ) = E SI λ) Ψ SI λ) at λ = 0. E By the argument used to prove A-3), SI λ) 0. So, the graph of E SIλ) lies above its secant lines. We notice that if ) Z > , then E HF = Z + Z < Z = E SI ), A-6) since H SI ) is a Hamiltonian with delta well potential of strength Z. Also, E d 0) = γ 0 < γ 0 = E SI 0) < 0. A-7) So E SI λ) lies above the secant line through 0, E SI 0) ) and, E SI ) ), and from A-6) and A-7), we see that this secant line lies above Lλ) for Z > + 3 ) Thus, if Z > + 3 ) , then E d λ) Lλ) < E SI λ) for all 0 λ. See the sketch on Figure 4. For λ < 0 we make the following observation: H d λ) = ) x + x Z [ δx ) + δx ) ] + φ 0x ) + φ 0x ) + λ [ δx x ) φ 0x ) + φ 0x ) ) ] = x x Z δx ) + λ) φ 0x ) Z δx ) + λ) φ 0x ) + λ δx x ) = H x ) SI λ) + H x ) SI λ) + λ δx x ) H x ) SI λ) + H x ) SI λ), A-8) where H x ) SI λ) = x Z δx ) + λ) φ 0x ) is the singly ionized Hamiltonian of an electron with spatial coordinate x and likewise for H x ) SI λ). This implies that for λ < 0, we have E d λ) < E SI λ) everywhere that E d λ) and E SI λ) exist. So E d λ) must remain a positive distance from the singly ionized threshold for λ, if Z > + 3 )

21 The electron electron bound state threshold We now show that Lλ) < E ee λ) = λ for λ < 0, provided that Z > Recall that E d λ) must lie on or below Lλ). So, whenever Z > and λ < 0, E d λ) Lλ) L ) = Z + 3 Z 5 < 4 E eeλ), A-9) Thus, provided Z is above this value we know that the bound state eigenvalue stays below this threshold for λ < 0. Refer to the sketch in Figure 5. An intersection between E ee λ) and E d λ) must occur further to the left than the intersection between E ee λ) and Lλ). Clearly, E d λ) Lλ) < 0 on λ for Z above this value, so Theorem 3. is proved. Proof of Theorem 3. The inequality A-3) is again satisfied by E ph λ), so we have E ph λ) 0, and the graph of E ph λ) lies below its tangent lines. We again let Lλ) be the tangent line at λ = 0 going through the points 0, E ph 0) ) and, E HF ). Since we do not know Ψ ph 0), we cannot use the same argument as in the delta function model. For now we note that since Ψ ph 0) minimizes Ψ, H ph ) Ψ over all Slater determinants, we have E HF = E ph 0) + E ph 0) = Ψ ph0), H ph ) Ψ ph 0) A-0) φx ) φx ), H ph ) φx ) φx ), for any normalized φ with φx ) φx ) in the operator domain of H ph ). The singly ionized threshold Let E SI λ) be the smallest eigenvalue of H SI λ). Again from A-3), we know that the graph of E SI λ) lies above its secant lines. Consider the trial function φ T x) = Z 5 ) 3/ [ exp Z 5 ) ] x. π 6 6

22 Then for Z > ).06694, E HF E T = φ T x ) φ T x ), H ph ) φ T x ) φ T x ) = Z 5 6 ) Z A-) = E SI ), A-) since H SI ) is the hydrogen atom Hamiltonian. With 3.4) and 3.7) we obtain A-7) for this model. The graph of E SI lies above the secant line through 0, E SI 0) ) and, E SI ) ). From A-) and A-7), we see that this line lies above Lλ) for Z > ). Thus, E ph λ) < E SI λ) on 0 λ for Z > ) For λ < 0 we use the same argument as in A-8). It follows that E ph λ) must remain a positive distance from E SI λ) for λ, provided Z > ) The electron electron bound state threshold As in the delta function model, there is a threshold at E ee λ) = λ 4, if Reλ) < 0. We prove below that whenever Z > 6 E ph 0) E ph 0) < 4, ).85. From this it follows that A-3) E ph λ) Lλ) < 4 E eeλ), A-4) for λ < 0. To prove A-3), we first note that 3.3) and 3.5) imply Ψ ph 0), x x Ψ ph0) = Ψ ph 0), νi HF Ψ ph 0) = φ 0 x i ), ν HF i φ 0 x i ).

23 Then, using 3.5) and A-), we have E ph 0) = = = φ 0, φ 0, φ 0, inf Z ) φ 0 x Z ) φ 0 x φ 0 x ) φ 0 x ), { σ Z ) φ 0 x ) } Z x + φ 0 x ), ν HF φ 0 x ) ) x x νhf ν HF E ph 0) E ph 0) φ 0 x ) φ 0 x ) = Z E ph 0) A-5) From A-), we know that E ph 0) + E ph 0) Z 5 ). Combining this with 6 A-5), we obtain E ph 0) E ph 0) Z Z This implies the bound A-3). The threshold at 0 is clearly not an issue, so Theorem 3. is proved. Acknowledgement It is a pleasure to thank Professor T. Daniel Crawford for his encouragement and many useful comments. References [] C. Møller and M. S. Plesset, Phys. Rev. 46, ). [] P. J. Knowles, K. Somasundram, N. C. Handy, and K. Hirao, Chem. Phys. Lett. 3, 8 985). [3] N. C. Handy, P. J. Knowles, and K. Somasundram, Theor. Chim. Acta. 68, ). [4] J. Olsen, O. Christiansen, H. Koch and P. Jørgensen, J. Chem. Phys. 05, ). [5] O. Christiansen, J. Olsen, P. Jørgensen, H. Koch, P. A. Malmqvist, Chem. Phys. Lett. 6, ). 3

24 [6] H. Larsen, A. Halkier, J. Olsen, and P. Jørgensen, J. Chem. Phys., ). [7] M. Leininger, W. D. Allen, H. F. Schaefer III, and C. D. Sherrill, J. Chem. Phys., ). [8] F. H. Stillinger, J. Cem. Phys., ). [9] D. Z. Goodson, Int. J. Quantum Chem. 9, ). [0] D. Z. Goodson and A. V. Sergeev, Adv. Quantum Chem. 47, ). [] A. V. Sergeev, D. Z. Goodson, S. E. Wheeler, and W. D. Allen, J. Chem. Phys. 3, ). [] A. V. Sergeev and D. Z. Goodson, J. Chem. Phys. 4, ). [3] J. P. Finley, J. Chem. Phys., ). [4] Y. K. Choe, H. A. Witek, J. P. Finley, K. Hirao, J. Chem. Phys. 4, ). [5] H. A. Witek, Y. K. Choe, J. P. Finley, K. Hirao, J. Comp. Chem. 3, ). [6] K. Yokoyama, H. Nakano, K. Hirao, J. P. Finley, Theor. Chem. Acct. 0, ). [7] T. W. Gamelin, Complex Analysis New York, Springer-Verlag New York, 00) [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators New York, London, Academic Press, 978). [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis New York, London, Academic Press, 97). [0] Y. Nogami, M. Vallières, and W. van Dijk, Am. J. Phys. 44, ). [] E. H. Lieb and B. Simon, Commun. Math. Phys. 53, ). [] M. Klaus and B. Simon, Ann. Phys. 30, ). [3] M. Klaus and B. Simon, Commun. Math. Phys. 78, ). 4

25 Im E O O OOX Re E X X X Figure : The spectrum of Hλ) in example, for λ =. The O are the eigenvalues of 0 Hλ) that accumulate at the lowest of the eigenvalues of x + λ) which are x labeled X. The lowest X is the bottom of the continuous spectrum, which is the filled half-line on the real axis. o Im E o oo X o o 0. o X XX Re E Figure : The spectrum of Hλ) in example, for λ = i. The O are the eigenvalues 0 of Hλ) that accumulate at the eigenvalues of x +λ) labeled X. Furthermore, x these values X are left-endpoints of half-lines that make up σ c Hλ) ). 5

26 Im E X O X O XXX OO Re E Figure 3: A qualitative sketch of the structure of σ c H d λ)) at some λ R and some Z > 0. The O are the eigenvalues of H SI λ) and the X are the eigenvalues of H ee λ) Λ Figure 4: A qualitative sketch of the eigenvalue and singly ionized threshold in the delta model, for 0.5 λ and some Z > 0. E d λ) is the thin solid curve, its tangent line at λ = 0 is the dashed line, E SI λ) is the thick curve, and the secant line through 0, E SI 0) ) and, E SI ) ) is dash dotted. We show that for appropriate Z and 0 λ, the graph of E d λ) lies below the tangent line, the tangent line lies below the secant line, and the secant line lies below the graph of E SI λ) Λ Figure 5: A qualitative sketch of the eigenvalue and electron-electron bound state threshold in the delta model, for.5 λ 0.5 and some Z > 0. E d λ) is the thin solid curve, its tangent line at λ = 0 is the dashed line, and E ee λ) is the thick curve. We show that for appropriate Z and λ < 0, the graph of E d λ) lies below the tangent line and the tangent line lies below the graph of E ee λ). 6

27 0.5 0 λ E d λ) E SI λ) Lλ) E SI secant line E ee λ) = λ / Figure 6: A qualitative sketch of the eigenvalue and thresholds in question in the delta model. Eλ) Figure 7: A numerical plot of the smallest six eigenvalues of h d λ) on λ, for Z =.38. λ 7

28 4 x 3 0 x Figure 8: A contour plot of the eigenfunction squared corresponding to the smallest eigenvalue of h d λ) at λ = 0.5, for Z = x 3 0 x Figure 9: A contour plot of the eigenfunction squared corresponding to the smallest eigenvalue of h d λ) at λ =.85, for Z =.38. 8

29 x x Figure 0: A contour plot of the eigenfunction squared corresponding to the second smallest eigenvalue of h d λ) at λ = 0, for Z = ReE) ImE) X eigenvalues of "attracting electron" states O eigenvalues of "singly ionized" states Figure : A qualitative sketch of the structure of σ c H ph λ)) at some λ = α + iβ, β 0 and some Z > 0. 9

30 ImE) ReE) Figure : A plot of the 45 eigenvalues with smallest real part of h d λ) at λ = i, for Z =.38. The o is λ which is the exact value of E 4 ee λ). Imλ) Reλ) Figure 3: A contour plot of the real part of the eigenvalue of smallest real part of h d λ) on the complex plane of λ, for Z=

31 Imλ) Reλ) Figure 4: A contour plot of the imaginary part of the eigenvalue of smallest real part of h d λ) on the complex plane of λ, for Z = Imλ) Reλ) Figure 5: A contour plot of the real part of the eigenvalue of smallest real part of h SI λ) in the delta model, on the complex plane of λ, for Z =

32 Imλ) Reλ) Figure 6: A contour plot of the imaginary part of the eigenvalue of smallest real part of h SI λ) in the delta model, on the complex plane of λ, for Z =.38. 3