An Improved Lower Bound to the Ground-State Energy - Journal of

Subscriber access provided by Washington University | Libraries

Letter

An Improved Lower Bound to the Ground State Energy Eli Pollak J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.9b00128 • Publication Date (Web): 12 Feb 2019 Downloaded from http://pubs.acs.org on February 13, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

An Improved Lower Bound to the Ground State Energy Eli Pollak∗ Chemical and Biological Physics Department, Weizmann Institute of Science, 76100, Rehovot, Israel E-mail: [email protected]

1

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract The Arnoldi iterative method for determining eigenvalues is based on the observation that the effect of operating with the Hamiltonian on a vector may be expressed as a sum of parallel and perpendicular contributions. This identity is used here to improve the previous lower bound estimate of the ground state energy by Temple, derived ninety years ago (Proc. Roy. Soc. (London) 1928 A119 276). The significantly improved lower bound is exemplified by considering a quartic and a Morse potential. The lower bound is valid for any Hermitian operator whose discrete spectrum is bounded from below.

Table of Content Graphic

2


Page 2 of 14

Page 3 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The Rayleigh Ritz variational principle is basic to quantum chemistry. The mean of the Hamiltonian using any normalized wavefunction is an upper bound to the true ground state energy ε0 . As it now stands though, the upper bound alone does not tell us how good it is. Variational optimization and convergence criteria can give a good feeling for the accuracy of the upper bound, but not a quantitative measure of its accuracy. A lower bound, with the same quality as the upper bound would provide objective knowledge on the accuracy of the estimate. It can also be used to improve it by the simple exercise of taking the average of the upper and lower bounds. A variational lower bound opens an alternative route for obtaining accurate estimates of the ground state energy. Finding a lower bound turned out though to be a challenging task. The seminal paper on this topic was published 90 years ago by Temple. 1 Additional milestones are the papers by Weinstein 2,3 and Delves. 4 In Weinstein’s formulation the standard deviation of the energy q 2 ) provides a lower bound to the ground state energy (ε ≥ ˆ 2 |Φi − hΦ|H|Φi ˆ ( σΦ = hΦ|H 0 ˆ ˆ hΦ|H|Φi− σΦ ) (provided that hΦ|H|Φi ≤ (0 + 1 )/2 with 1 denoting the exact first excited state energy). In practice, due to the linear dependence on the standard deviation σΦ , this ˆ lower bound is not very tight. Temple’s result (ε0 ≥ hΦ|H|Φi −

2 σΦ ) ¯1 −hΦ|H|Φi ˆ E

(where E¯1 is

an estimate of the first excited state energy) improves the tightness significantly but even after 90 years, the lower bound is usually not as close to the true result as the Rayleigh-Ritz upper bound. 5–7 Attempts have been made to improve Temple’s lower bound. 5,8 One approach that has been studied rather extensively is based on finding variational upper bounds to a suitably defined inverse of the Hamiltonian operator. 9–12 Temple’s formula is for a pre-specified wavefunction. The variational procedure optimizes this wavefunction such that for a given basis set it leads to the ”best” Temple lower bound. 11 Marmorino and coworkers 6,11,13,14 have discussed various scenarios, perhaps the most relevant one in the context of this letter 6 shows that an improved bound may be obtained by including the third moment of the Hamiltonian. Yet this lower bound is implicit, not explicit. In this letter we derive a new explicit

3



Page 4 of 14

and significant improvement to Temple’s lower bound. ˆ denote the Hamiltonian operator (or any other Hermitian operator for that matter) Let H ˆ n i = εn |ϕn i. which has a complete set of eigenvalues and (normalized) eigenfunctions H|ϕ We assume that with some methodology or other we have a (normalized) state |Ψ0 i which ˆ 0 i bounds is a reasonable approximation to the ground state. The energy E0 = hΨ0 |H|Ψ the true ground state energy ε0 from above. The standard deviation of the estimate for q ˆ 2 |Ψ0 i − hΨ0 |H|Ψ ˆ 0 i2 is assumed to be known. As also the ground state energy σ0 = hΨ0 |H considered expressly below, knowledge of the standard deviation and the third moment of the energy, allows for an improved estimate (λ0 ) of the ground state energy, relative to E0 . Our point of departure is the iterative Arnoldi method of determining eigenvalues. 15 The method depends on the observation that the operation of the Hamiltonian operator on the state |Ψ0 i can be represented as ˆ 0 i = E0 |Ψ0 i + σ0 |Ψ1 i H|Ψ

(1)

where |Ψ1 i is a (normalized) state perpendicular to the state |Ψ0 i, that is hΨ0 |Ψ1 i = 0. This identity is verified by multiplying from the left by hΨ0 | to see that the coefficient for ˆ 0 i. Similarly, multiplying from the left by the state |Ψ0 i on the right side is indeed hΨ0 |H|Ψ ˆ verifies the coefficient σ0 for the perpendicular contribution on the right hand side of hΨ0 |H the identity. The normalized state |Ψ1 i is thus

|Ψ1 i =

i 1 hˆ H − E0 Iˆ |Ψ0 i. σ0

(2)

It has the additional property that

ˆ 0 i = σ0 hΨ1 |H|Ψ

as may be seen by multiplying Eq. 1 from the left by hΨ1 |.

4


(3)

Page 5 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Using the representation of the Hamiltonian matrix with the two states |Ψ0 i and |Ψ1 i, denoting ˆ 1i E1 = hΨ1 |H|Ψ

(4)

(we are assuming without loss of generality that E1 ≥ E0 ) and diagonalizing one finds (as mentioned above) an improved estimate for the ground state energy denoted by λ0 as q 1 2 2 E0 ≥ λ 0 = E0 + E1 − (E1 − E0 ) + 4σ0 ≥ ε0 . 2

(5)

ˆ 0 i (see Eq. 2), the information needed to establish Since the state |Ψ1 i has a contribution H|Ψ the energy E1 includes the third moment of the energy. This improved estimate is thus based on the second and third moments of the energy using the state |Ψ0 i. At this point it is instructive to re-derive Temple’s lower bound. Multiplying Eq. 1 from the left by the true ground state of the Hamiltonian and squaring gives the result σ02 |hϕ0 |Ψ1 i|2 [E0 − ε0 ] = . |hϕ0 |Ψ0 i|2 2

(6)

By construction, the identity operator in the Hilbert space of the Hamiltonian may be written as: IˆN = |Ψ0 ihΨ0 | + |Ψ1 ihΨ1 | +

∞ X

ˆ |Ψj ihΨj | ≡ |Ψ0 ihΨ0 | + |Ψ1 ihΨ1 | + Q

(7)

j=2

where the states |Ψj i, j = 2, .., are orthonormal to each other as well as |Ψ0 i and |Ψ1 i, and ˆ Eq. 7 implies that the second equality on the r.h.s defines the projection operator Q.

ˆ 0 i. |hϕ0 |Ψ1 i|2 = 1 − |hϕ0 |Ψ0 i|2 − hϕ0 |Q|ϕ

(8)

Inserting this equality into Eq. 6, noting that E0 ≥ ε0 leads to the identity s ε 0 = E 0 − σ0

ˆ 0i 1 − |hϕ0 |Ψ0 i|2 − hϕ0 |Q|ϕ . 2 |hϕ0 |Ψ0 i| 5


(9)


Page 6 of 14

ˆ 0 i ≥ 0 we find the inequality Since by definition, hϕ0 |Q|ϕ r ε 0 ≥ E0 − σ 0

1 − a0 . a0

(10)

and we denoted |hϕ0 |Ψ0 i|2 ≡ a0 . It remains to obtain an estimate for the projection coefficient onto the ground state a0 . 16 Without loss of generality we may write ˆ 0 i ≡ a0 ε0 + (1 − a0 ) E¯1 hΨ0 |H|Ψ

(11)

ˆ 2 |Ψ0 i ≡ a0 ε2 + (1 − a0 ) E¯2 hΨ0 |H 0

(12)

where we used the notation P∞ E¯1 =

|hϕj |Ψ0 i|2 εj , E¯2 = (1 − a0 )

j=1

P∞

|hϕj |Ψ0 i|2 ε2j . (1 − a0 )

j=1

(13)

Although E¯1 and E¯2 are unknown this is not a major obstacle as we shall see. With some manipulation of Eqs. 11 and 12 one finds that

σ02

i2 (1 − a0 ) h ˆ ¯ hΨ0 |H|Ψ0 i − E1 + (1 − a0 ) E¯2 − E¯12 = a0 i2 (1 − a0 ) h ˆ 0 i − E¯1 ≥ hΨ0 |H|Ψ a0

(14)

Inserting this last inequality into Eq. 10 leads directly to the Temple lower bound expression σ2 ε 0 ≥ ET ≡ E0 − ¯ 0 . E1 − E0

(15)

Formally, this is an improvement on Temple’s result, since here the meaning of the excited state energy E¯1 is precisely defined through Eq. 13 and bounds the first excited state energy from above. In practice, the unknown excited state energy E¯1 is estimated from some

6


Page 7 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


knowledge on the energy of the first excited state since as may be inferred from Eq. 13, E¯1 ≥ ε1 . The Temple lower bound was improved by Marmorino and Gupta 6 by employing tighter bounds to the coefficient a0 . The aim of this letter is to go beyond the Temple lower bound. The key is the identity given in Eq. 9. A better bound will be obtained if we can estimate matrix elements of the ˆ For this purpose we formally construct a normalized state |Ψ2 i such projection operator Q. that it is perpendicular to both |Ψ1 i and |Ψ0 i: 1

|Ψ2 i = p

h

σ12 + σ02

i ˆ − E1 Iˆ |Ψ1 i − σ0 |Ψ0 i . H

(16)

q ˆ 2 |Ψ1 i − hΨ1 |H|Ψ ˆ 1 i2 . It is then a matter of straightforward algebra to where σ1 = hΨ1 |H see that |hϕ0 |Ψ2 i|2 =

1 2 2 (E − ε ) (E − ε ) − σ |hϕ0 |Ψ0 i|2 1 0 0 0 0 2 2 2 σ0 [σ0 + σ1 ]

(17)

This may be inserted into the identity given in Eq. 9 to improve the Temple lower bound by writing: 2

[E0 − ε0 ]2 ≤ σ02

1 − a0 [σ02 − (E1 − ε0 ) (E0 − ε0 )] − . a0 [σ02 + σ12 ]

(18)

Of course we do not know the ground state energy ε0 , so a bit of additional algebra is needed. Denoting X = λ0 − ε0 ≥ 0 (the improved estimate for the ground state energy λ0 has been given in Eq. 5), noting the eigenvalue equation for λ0 ((E1 − λ0 )(E0 − λ0 ) = σ02 ), we find the intermediate relation:

2

[E0 − λ0 + X]

≤ ≤

2 − a0 2 [(E1 − λ0 + E0 − λ0 ) + X] −X a0 [σ02 + σ12 ] 2 σ04 2 [(E1 − λ0 + E0 − λ0 )] − X 2 [σ02 + σ12 ] E¯1 − E0

1 σ02

(19)

where in the second line we used the Temple inequality for the first term and the fact that X ≥ 0 in the second term. Since X 2 and Temple’s lower bound squared scale at least as σ04

7



Page 8 of 14

it is clear from the second term on the right hand side of Eq. 19, that like Temple’s bound, the improved lower bound will scale at least as σ02 . It is then a matter of straightforward manipulation to find the central result of this letter: (E0 − λ0 ) ε0 ≥ E< ≡ λ0 − [E0 +E1 −2λ0 ]2 1+ [σ02 +σ12 ] v  h i u 2 u [E0 + E1 − 2λ0 ]2 (E1 − λ0 )2 − E¯1 − E0 u (E1 − λ0 )2  t · − 1 2 + 2   . (20) E¯1 − E0 [σ02 + σ12 ] E¯1 − E0

In practice, as for the Temple lower bound, one needs an estimate for the excited state energy E¯1 . Since E¯1 ≥ 1 one way of testing the quality of the improved bound is by replacing E¯1 by the first excited state energy 1 (this is the strategy used in the numerics below) as also done for example, in Ref. 7 when assessing Temple’s lower bound. In comparison with the Temple lower bound, the added information for this improved ˆ 3 |Ψ0 i and hΨ0 |H ˆ 4 |Ψ0 i which is needed to determine E1 bound (E< ) is the knowledge of hΨ0 |H and σ1 . The energy E1 depends on the third moment so the only added information relative to the upper bound estimate λ0 is the fourth moment. This is analogous to the Temple bound, where to obtain the upper bound E0 one needs the first moment of the energy while to obtain the Temple lower bound one needs in addition the second moment. The critical reader might comment that one may apply the Temple lower bound to the improved eigenfunction obtained from diagonalization of the two dimensional Hamiltonian matrix which gives the improved upper bound λ0 . This normalized eigenfunction is:

|Ψλ0 i =

[σ0 |Ψ0 i − (E0 − λ0 ) |Ψ1 i] q σ02 + (E0 − λ0 )2

(21)

ˆ λ0 i. It is straightand obviously the improved ground state energy estimate λ0 = hΨλ0 |H|Ψ forward to find the second moment and then apply the Temple lower bound. However, as

8


Page 9 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


also verified through numerical computation, this does not provide an essential improvement as compared to the original Temple lower bound given in Eq. 15 since it does not include ˆ higher order terms coming from the projection operator Q. It remains to demonstrate the quality of the improved lower bound of Eq. 20. For this purpose we study two model systems. One is the quartic oscillator where the spacing between eigenstates increases with energy, the other is the Morse oscillator where the spacing decreases with increasing energy. The quartic Hamiltonian we use is

HQ = −

1 1 d2 + x4 . 2 2 dx 2

(22)

The lowest symmetric eigenvalues have been reported in Ref. 17 as .530181 and 3.72785. As 1/4 a trial wavefunction we use the Gaussian Ψ0 (x) = Γπ exp (−Γx2 /2) where the width parameter Γ will serve as the variational one. The necessary Gaussian integrals are readily performed using Maple or Mathematica so we do not specify them here, but rather present the results. In panel (a) of Fig. 1 we show the upper bounds E0 (Γ) and λ0 (Γ) and the lower bounds ET (Γ) and E< (Γ) as functions of the variational parameter Γ. Note the improvement in the lower bound E< (Γ) as compared to the Temple lower bound ET (Γ) (both obtained using the excited state energy 3.72785 for E¯1 ). To see this in more detail we plot in panel (b) of the Figure the gap between the upper and lower bounds and the numerically exact eigenvalue. The best upper (lower) bound is found at Γ = 1.40647 (1.88026) and the gap is 0.0019675 (0.0029643). The quality of both bounds is similar. As a comparison, the best Temple lower bound found at Γ = 1.87128 has a gap of 0.0057772, it is twice as large as the improved lower bound gap and almost three times the upper bound gap. The second example is a Morse oscillator potential

HM = −

1 d2 + 8 [exp(−x) − 1]2 2 2 dx

9


(23)


Figure 1: Upper and lower bounds to the ground state energy of a quartic oscillator. Panel (a) shows the variational estimate E0 (Γ) (upper blue dotted line) the improved variational estimate λ0 (Γ) (upper red dashed line), the numerical ground state energy (black solid line), the improved lower bound of this letter (lower red dashed line) and Temple’s lower bound estimate (lower blue dotted line) as a function of the width parameter Γ of the initial Gaussian wavefunction. Panel b shows the gap between the improved upper bound λ0 (Γ) and the exact ground state energy (solid blue line) and the gap between the ground state energy and the improved lower bound E< (Γ) of this letter (dashed red line) as a function of Γ. The quality of these upper and lower bounds is similar. chosen such that it supports 7 bound states, or in other words it is highly anharmonic. The ground and first excited state energies are 1.875 and 4.875 respectively. The wavefunction 1/4 exp −Γ (x − a)2 /2 is chosen as a shifted Gaussian, due to the asymmetry Ψ0 (x) = Γπ of the potential and the width (Γ) and location (a) are the variational parameters. The minimum of the variational energy E0 (Γ = 3.742, a = 0.2004) is 1.93611 corresponding to a gap of ∆E0 = 0.0611. The best Temple lower bound (using the excited state energy of 4.875) is ET (Γ = 5.032, a = 0.0724) = 1.7361 corresponding to a Temple gap of ∆ET = 0.1389. The improved optimized upper bound is λ0 (Γ = 3.6543, a = 0.2025) = 1.9012 with a gap of ∆λ = 0.02622. The best lower bound found using Eq. 20 is E< (Γ = 5.023 =, a = 0.0712) = 1.7858 and the corresponding gap ∆E< = 0.0892 is 50% smaller than the Temple gap. In summary, a significantly improved lower bound to the ground state energy has been 10


Page 10 of 14

Page 11 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


derived and applied to a quartic and a Morse potential. For the Temple lower bound one needs to know the first and second moments of the energy as well as an estimate for the first excited state energy. For the improved bound one needs to know in addition the third and fourth moments of the energy. The improved lower bound is based on a wavefunction approach and so is naturally applicable to computations which use the variational principle and especially to Krylov space 18 based computations which in any case generate higher moments of the Hamiltonian. Along the way, a new improved and rather straightforward derivation of the Temple lower bound has been given. Obtaining the improved lower bound estimate does not necessitate a full diagonalization of the Hamiltonian which is costly and scales as the dimensionality cubed of the basis set. One may use iterative methods 19 which scale as the dimensionality squared and provide an estimate of the first few moments of the Hamiltonian. It is important to stress that the input needed for the improved lower bound is based on the moments and the estimate of the excited state energy only, there is no need in practice for wavefunction information. The methodology considered here can be systematically improved. It is not necessary to stop with only three orthonormal wavefunctions. Construction of a fourth wavefunction implies a straightforward generalization of Eq. 16. By construction, its use guarantees a further improvement of the lower and upper bounds. Here too it should be stressed that the wavefunction construction is formal, the input needed will be only higher order moments of the Hamiltonian. The central identity of this letter, Eq. 9 was written down for the ground state energy, but it is equally applicable to any eigenvalue of the Hamiltonian with the obvious modifications. Eq. 9 would lead to a bound on the difference between any of the computed and exact eigenvalues. Will the improved lower bound presented here become a staple of eigenvalue computation? At this point we can only speculate. We know for example that the Temple lower bound for the ground state of the Li atom is four orders of magnitude less tight than the

11



respective upper bound based on using the same basis states. 10 The same is true for the He atom ground state energy. 7 It is a daunting task to reduce this large difference. On the other hand, the computation for the He atom is based on 300 basis functions and not only three basis functions as used in the examples for the quartic and Morse oscillator potentials presented in this letter. A systematic application of the present methodology, using an increasing number of Arnoldi basis states to improve the assessment of the contribution of the projection operator (Eq. 9 should considerably tighten the lower bound. The improved lower bound is not only to be considered in the context of electronic energies. For example, lower bound formulae have been employed in the context of condensed matter systems. 20 The variational upper bound principle is a staple of the density-matrix renormalization group (DMRG) methodology used to find ground state energies of strongly coupled quantum systems 21 which also employs Arnoldi type basis sets. It should be interesting to study the present generalization of the Temple lower bound and its applicability within the DMRG methodology. Finally, we note that in this letter we considered the Hamiltonian operator but the derivation remains the same for any Hermitian operator whose discrete spectrum is bounded from below.

Acknowledgements I thank Professor Salvador Miret-Art’es for his constructive and helpful comments on various versions of this letter. This work was generously supported by grants of the Israel Science Foundation and the Minerva Foundation, Munich.

References (1) Temple, G.; The Theory of Rayleigh’s Principle as Applied to Continuous Systems Proc. Roy. Soc. (London) 1928 A119 276 12


Page 12 of 14

Page 13 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(2) Weinstein, D.H.; Modified Ritz Method, Proc. Nat. Acad. Sci. USA 1934 20 529-532 (3) Stevenson, A.F.; On the Lower Bounds of Weinstein and Romberg in Quantum Mechanics, Phys. Rev. 1938 53 199 (4) Delves, L.; On the Temple Lower Bound for Eigenvalues, J. Phys. A: Gen. Phys. 1972 5 1123-1130 (5) Hill, R.N.; Tight Lower Bounds to Eigenvalues of the Schr¨odinger Equation, J. Math. Phys. 1980 21 2182-2192 (6) Marmorino, M.G.; Gupta, P.; Surpassing the Temple Lower Bound, J. Math. Chem. 2004, 35 189-197 (7) Donchev, A.G.; Kalachev, S.A.; Kolesnikov, N.N.; Tarasov, V.I.; The Upper and Lower Bounds of Energy for Nuclear and Coulomb Few-Body Systems, Phys. Part. Nucl. Lett. 2007 4 39-45 (8) Cohen, M; Feldmann, T.; Lower Bounds to Eigenvalues, Can. J. Phys. 1969 47 18771879 (9) Scrinzi, A.; Lower Bounds to the Binding Energies of tdµ, Phys. Rev. A 1992, 45 7787-7791 (10) L¨ uchow, A.; Kleindienst, H.; Accurate Upper and Lower Bounds to the 2 S States of the Lithium Atom, Int. J. Quant. Chem. 1994, 51 211-224 (11) Marmorimo, M.G.; Equivalence of Two Lower Bound Methods, J. Math. Chem. 2002, 31 197-203 (12) Toth, S.; Szabados, A.; Energy Error Bars in Direct Configuration Interaction Iteration Sequence, 2015, 143 085112(1-7)

13



(13) Marmorino, M.G.; Almayouf, A.; Krause, T.; Le, D.; Optimization of the Temple Lower Bound, J. Math. Chem. 2012, 50 833-842 (14) Marmorino, M.G.; Black, V.; Lower Bounds to the Ground-State Expectation Value of Non-negative Operators, J. Math. Chem. 2016, 54 1973-1985 (15) Arnoldi, W.E.; The Principle of Minimized Iterations in the Solution of the Matrix Eigenvalue Problem, Quarterly of Appl. Math. 1951 9 17-29 (16) Weinhold, F.; Criteria of Accuracy of Approximate Wavefunctions, J. Math. Phys. 1970 11 2127-2138 (17) Reid, C.E.; Energy Eigenvalues and Matrix Elements for the Quartic Oscillator, J. Molec. Spect. (1970 36 183-191. (18) Krylov, A.N.; On the Numerical Solution of Equation by Which are Determined in Technical Problems the Frequencies of Small Vibrations of Material Systems, Otdel. mat. i estest. nauk, 1931 7 491-539 (in Russian) (19) Davidson, E.R.; The Iterative Calculation of a Few of the Lowest Eigenvalues and Corresponding Eigenvectors of Large Real-Symmetric Matrices, J. Comp. Phys. 1975 17 87-94. (20) Baumgratz, T.; Plenio, M.B.; Lower Bounds for Ground States of Condensed Matter Systems, New J. Phys. 2012 14 023027 (1-24) (21) Schollw¨ock, U.; The Density Matrix Renormalization Group in the Age of Matrix Product States, Ann. Phys. 2011 326 96-192

14


Page 14 of 14

An Improved Lower Bound to the Ground-State Energy - Journal of

Recommend Documents