About the Variational Property of Generalized Discrete Variable

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About the Variational Property of Generalized Discrete Variable Representation Viktor Szalay J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp400112p • Publication Date (Web): 25 Mar 2013 Downloaded from http://pubs.acs.org on March 26, 2013

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The Journal of Physical Chemistry

About the Variational Property of Generalized Discrete Variable Representation Viktor Szalay1, a) Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences. P. O. Box 49, H-1525 Budapest, Hungary (Dated: 25 March 2013)

It is explained why the generalized discrete variable representation is variational neither with respect to the size (truncation) of the basis set nor with respect to the selection of grid points. Keywords: matrix representation, Schrödinger equation, molecular vibrations, effective operators

a)

Electronic mail: [email protected]

Typeset by REVTEXand AIP

1

ACS Paragon Plus Environment

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I.

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INTRODUCTION Rotational-vibrational spectroscopy gives a wealth of information about the structure

and nuclear motion dynamics of molecules. Theoretical modeling of rovibrational spectra is instrumental in understanding the often complicated experimental spectra. The rovibrational levels for semirigid polyatomic molecules can be calculated very efficiently from first principles by the code MULTIMODE1 employed in conjunction with the ingenious potential energy surface construction method developed by Braams and Bowman2 . First principles calculation of rovibrational levels of floppy polyatomic molecules consisting of as many as up to six atoms3–11,15–17 may be carried out efficiently by employing the discrete variable representation (DVR) method12–14 . The present communication is concerned with the variational properties of the DVR. It is interesting to note that variational properties of a purely grid based method, the finite difference (FD) method has been investigated in electronic structure calculations18,19 . The FD method is often employed in density-functional theory calculations and it is well known that it gives nonvariational results as the grid changes20 . Researchers in this field have made efforts to understand why FD is nonvariational and to obtain an FD which is variational with respect to the grid18,19 . Since the DVR is pseudospectral method employing a truncated basis set and a set of grid points (non)variational behaviour both with respect to basis truncation and to changes of grid points must be investigated. We shall explain by means of analytical derivations why the DVR is variational neither with respect to the size of the basis set nor with respect to the selection of grid points. The motivation, basic ideas and outline of the paper is as follows. The hallmark of DVR is the diagonal potential energy matrix, Vαβ = V (qα ) δαβ ,

(1)

where Greek subscripts number the grid points and qα denotes the αth point qα = (q1α , q2α , . . . , qDα ) in D-dimensional configuration space of the system studied. By employing similarity transformation the DVR can be transformed into an equivalent representation, the finite basis representation (FBR)13 . The transformation involves a set of basis functions and the grid points. The number of basis functions and grid points is equal. In essence, the FBR is a representation where the exact matrix elements of the potential energy operator in a truncated basis (i.e. the variational basis representation (VBR)) are approximated by numerical quadrature employing the DVR grid. To obtain DVR(FBR) of a Hamiltonian 2


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operator of reasonable accuracy one must ensure, as a minimum requirement, that the numerical quadrature is exact for the overlap matrix of the basis functions. Within the generalized DVR21,22 it is ensured for nearly all selection of grid points. This warrants the name generalized DVR. The generalized DVR includes as a special case the standard DVR utilizing Gaussian-quadrature grid. The generalized DVR allows one to use many different sets of grid points. The different sets give results, i.e. sets of eigenvalues, of different accuracies. The question arises that how one could find the set of grid points leading to the possible highest accuracy23 . As demonstrated by numerical examples23 , with N grid points optimized for eigenvalues (energy) the generalized FBR(DVR) of N -basis functions can give results of orders of magnitude higher accuracy than the related VBR calculations employing the same N basis functions. Were the DVR variational with respect to the grid energy optimized grids could be found with ease. Numerical experience shows, however, that the generalized DVR is not variational with respect to the grid either24 . The explanation for the DVR(FBR), generalized or standard, not having variational property may appear simple: There is at least one potential energy matrix element in the FBR which is not calculated (numerically) exactly by the quadrature. Similary simple is the idea of restoring the variational property: To obtain variational behaviour one must increase the accuracy of the quadrature by increasing the number of grid points. One might have objections, however: First, by increasing the number of grid points to approximate the VBR matrix elements as accurately as possible one gives up the DVR idea13 of using as many grid points as the dimension of the subspace of the Hilbert space where solution to the eigenvalue equation of the Hamiltonian operator is looked for. Second, no matter how many grid points are employed the grid points are not variational parameters. Indeed, while keeping the number of grid points fixed but varying the grid points the calculated eigenvalues show no variational behaviour24 . That is the explanation given above fails to explain the nonvariational behaviour with respect to the grid, nor it gives any hint about how variational behaviour with respect to the grid might be obtained. It is worth summarizing: a) The FBR may be variational with respect to the basis set at the cost of using more grid points than the number of basis functions, and then, it is identical with the VBR apart from some small numerical error. b) Whether or not the FBR(DVR) is variational with respect to the basis set it is nonvariational with respect to the grid. This makes the calculation 3


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of the optimal set of grid points giving eigenvalues of the possible highest accuracy very difficult23 . If one wants to calculate eigenvalues with the possible highest accuracy one should optimize the grid points with respect to the eigenvalues, rather than optimizing and/or adding more grid points to obtain matrix elements of the possible highest accuracy. Clearly, DVR having variational property with respect to the grid could be very useful, for it would make calculation of energy (eigenvalue) optimzed grids, such as derived in ref 23, much simpler. It might seem that this is all one can tell about the variational property of the DVR(FBR). Recently it has been shown, however, that DVR(FBR) can be derived by calculating all matrix elements of a DVR(FBR) Hamiltonian operator analytically. Since no quadrature is employed in deriving the DVR(FBR), one cannot argue that nonvariational behaviour of the FBR(DVR) is a straightforward consequence of quadrature error. Furthermore, as shown analytically and demonstrated numerically24 , DVR(FBR) which is variational with respect to the basis set and the grid points can be obtained without having to increase the number of grid points. This is achieved by a special construction of FBR(DVR) Hamiltonian operators and by making the product approximation exact. In the product approximation the VBR of products of coordinate operators are approximated as products of the VBRs of the coordinate operators25 . The product approximation is related to the quadrature approximation in certain cases26,27 . Note, that errors due to the product approximation can give only partial account for the nonvariational behaviour of the DVR(FBR), since they fail to explain nonvariational behaviour with respect to the grid. Albeit it is nonvariational, the generalized FBR(DVR) of ref 21 can assume nearly any sets of grid points. Therefore, one wonders if it could be derived as nonvariational approximation to a variational generalized FBR(DVR), that is a variational DVR where there is large selection (nearly free choice) of DVR points (as opposed to the case of variational DVRs derived by assuming finite dimensional Hilbert space24 ). Such a derivation would constitute a proper explanation of why DVR is nonvariational with respect to the grid and truncation of the Hilbert space. We shall carry out this derivation by extending ideas of ref 24 to infinite dimensional Hilbert space. As a side result we shall obtain a new generalized DVR (Eq. (43)) and an expression of variational generalized DVR (Eq. (42)). Besides the role it plays in understanding nonvariational behaviour of FBR(DVR), the derivation of a variational generalized 4


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FBR(DVR) is of importance in its own right: The derivations of variational FBR(DVR) described in ref 24 assumed Hilbert space of finite dimension. However, the finiteness of the Hilbert space strongly restricts the set of grid points from which the FBR(DVR) grid can be selected. One would like to have an FBR(DVR) variational with respect to the grid such that there is a significantly larger pool of grid points from which the FBR(DVR) grid can be selected. This may be achieved by extending the results of ref 24 to Hilbert spaces of infinite dimensions. A large point set over which the DVR is variational, would allow more flexible choice of grid points to obtain higher accuracy and faster convergence, and would facilitate the incorporation of symmetry consideration into the calculations. Though the DVR given in Eq. (42) is variational by construction, since it contains Dirac’s delta functions, it is not useful to numerical calculations. Nevertheless, by finding variational approximations to Eq. (42) a variational generalized DVR suitable to numerical computations could be obtained. Our derivations proceed as follows. In the first step (Section II), we shall construct commuting modifications of the projected coordinate operators P qi P, i = 1, 2, ..D (denoted by P qP in brief), where P denotes the projection into a subspace of the Hilbert space where solutions of the Schrödinger equation are looked for. It turns out that the modified operators is of the form P (q + qω) P where ω is a reduced wave operator. Then, it is shown that q and ω obey the decoupling equation (Section III). This result is of importance for the satisfaction of the decoupling equation is a must to derive DVR of variational property24 . Since the decoupling equation is obeyed, P (q + qω) P are effective coordinate operators. To introduce effective coordinate operators into the potential energy a Hamiltonian operator H related to the original energy operator H = H0 + V (q)

(2)

by similarity transformation, H = e−ω Heω , is considered. In Section IV it is shown that H is quasi-Hermitian28 . Then, the variational property associated with quasi-Hermitian operators is exploited to obtain, through a number of steps of derivation, a DVR which is variational with respect to both the basis size and the selection of grid. This DVR consists of two terms. The first term is a representation of a non-Hermitian Hamiltonian operator, and it is recognized as the expression of the generalized DVR. Therefore, the generalized DVR cannot be variational unless the second term vansihes. We shall show that under no circumstances the second term can vanish. Finally, Section V summarizes the results. 5


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With the outline and basic ideas of the paper presented we shall turn to the details now.

II.

COMMUTING MODIFICATION OF PROJECTED COORDINATE

OPERATORS Usually one obtains the DVR grid points as the eigenvalues of a commuting set of projected coordinate operators P qP . Then the common set of eigenstates of P qP provide the transformation to the DVR. The grid points are fixed by P and the requirement of having commuting P qi P, i = 1, 2, ..D places a severe restriction on the subspace projeted by P . Namely, it has to have direct product structure. We want to get free of this restriction and to have more flexible choice of grid points. The αth eigenstate of the coordinate operator q = (q1 , q2 , . . . , qD ) satisfies the eigenvalue equation q|qα i = qα |qα i.

(3)

P q|qα i = qα P |qα i.

(4)

Thus with any P and qα ,

In general, qα are not eigenvalues and P |qα i are not eigenstates of the projected coordinate operators P qP . Nevertheless, as shown below, they are eigenpairs of a set of modified projected coordinate operators, P (q + qω) P . Exactly it is this result which will allow us to derive the diagonal potential energy representation, Eq. (1), with general sets of grid points. We shall rewrite P q|qα i by assuming that P |qα i, α = 1, 2, . . . , N , where N denotes the dimension of the subspace projected by P , are linearly independent states. Apart from requiring linear independence of P |qα i no other condition restricts the choice of grid points. The states orthonormal to P |qα i will be denoted as |q¯β i, β = 1, 2, . . . , N , i.e. hq¯β |P |qα i = δβα .

(5)

The identity operator I is decomposed as I = P + Q, 6


(6)

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with Q denoting the projection operator on the subspace complementary to the one projected by P in the Hilbert space. Observe that P q|qα i = (P qP ) P |qα i + (P qQ) Q|qα i X = (P qP ) P |qα i + (P qQ) Q|qβ iδβα β

= (P qP ) P |qα i +

X

(P qQ) Q|qβ ihq¯β |P |qα i

β

"

=P q+q

X

Q|qβ ihq¯β |P

β

!#

P |qα i,

(7)

That is [P (q + qω) P ] P |qα i = qα P |qα i

(8)

with ω=

X

Q|qβ ihq¯β |P.

(9)

β

One can see that the projected, modified coordinate operators P (qi + qi ω) P possess common eigenvectors. Therefore, these operators commute even when the operators P qi P do not. Their spectral decomposition is P (q + qω) P =

X

P |qα iqα hq¯α |P.

(10)

α

One can obtain the diagonal representation, Eq. (1), of the potential energy operator as follows: 1) Replace q in V (q) with s, where s denotes a set of coordinates such that P sP = P (q + qω)P . 2) Project the result of step 1) with P . 3) Invoke the product approximation, that is approximate P V (s)P with V (P sP ). 4) Substitute P sP by its spectral decomposition, Eq. (10). 5) Form matrix representation in the biorthonormal basis {P |q¯α i, P |qα i}. This final step gives Eq. (1). It is nice that DVR of the potential energy operator could be derived with minimal restriction on the grid points and no restriction on P . The problem is how to carry out steps 1) and 3) without introducing approximation and ensuring that the representation obtained for the Hamiltonian operator is variational with respect to truncation of the Hilbert space and selection of the grid points. 7


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Let s be defined by the similarity transformation s = e−ω qeω .

(11)

Then s can be introduced into the Hamiltonian operator H as H = e−ω Heω

(12)

= e−ω H0 eω + V (s) .

(13)

Since similarity transformation of H leaves its eigenvalues intact, H and H have the same eigenvalues. In Section IV it will be shown how a variational principle can be used to approximate the eigenvalues of H and to obtain a Hamiltonian operator, Eq. (30), acting in the subspace projected by P . This Hamiltonian operator is not only a variational approximation with respect to P (i.e. truncation of the Hilbert space) to the original Hamiltonian, but it is also variational with respect to the choice of ω. Since ω depends on the grid points, this Hamiltonian is in fact variational with respect to the selection of grid points as well. To preserve the variational properties during step 3) we must require that P V (s)P = V (P sP ). One may show by induction24 that the product approximation becomes exact if the equation, called the decoupling equation, QsP = 0 is obeyed by ω. Since we have defined ω in Eq. (9) with no reference to the decoupling equation, it is important to check if this ω obeys the decoupling equation. It will be checked in the next Section. In passing it is worth noting that the decoupling equation is so called, for if it is obeyed, interaction between the P and Q spaces is removed entirely, and the eigenvalues of P sP will be identical with some of the eigenvalues of q and its eigenstates will be related to those of q by the relation |qα i = P |qα i + ω|qα i. The operators P sP of this property are called effective coordinate operators.

III.

DOES ω SATISFY THE DECOUPLING EQUATION?

The decoupling equation can be rewritten as Q (q + qω) P − Qω (q + qω) P = 0.

(14)

By using the defining equation of ω, Eq. (9), along with the spectral decompostion of the operator, Eq. (10), and the orthogonality relationship Eq. (5) one finds that the second 8


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term in Eq. (14) is Qω (q + qω) P = QωP

X

P |qα iqα hq¯α |P

(15)

α

=

X

Q|qβ ihq¯β |P

X

P |qα iqα hq¯α |P

(16)

α

β

=

X

Q|qα iqα hq¯α |P.

(17)

α

The first term in Eq. (14) can be written as Q (q + qω) P = QqP + QqωP X = QqP + Qq Q|qα ihq¯α |P

(18) (19)

α

= QqP + Qq

X

|qα ihq¯α |P − QqP

α

=

X

P |qα ihq¯α |P

(20)

α

X

Q|qα iqα hq¯α |P,

(21)

α

where we have made use of Eq. (3) and the fact that the projection operator P can be written as P =

X

P |qα ihq¯α |P.

(22)

α

One can see that the two terms in Eq. (14) cancel, that is ω obeys the decoupling equation. Therefore, the projected modified coordinate operators are also effective coordinate operators.

IV.

THE GENERALIZED DVR AS AN APPROXIMATION TO A

VARIATIONAL DVR Since the transformation relating the Hamiltonian operators H and H is not unitary, the basis expansion and truncation method of solving the Schrödinger equation for H is not variational. Below it is shown that H is, however, quasi-Hermitian. Quasi-Hermiticity implies that the basis expansion and truncation method is variational for the ground state of H provided that all matrix elements are calculated exactly. Then, this allows us to formulate a DVR for H which is variational for the ground state with respect to both the trunction of the basis and the selection of grid points. 9


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Recall the definition of quasi-Hermiticity28 : H is quasi-Hermitian, if there exists a positive definite operator T , a metric operator, acting in the Hilbert space, such that T = T †,

(23)

where supescript † denotes the adjoint, and the equation T H = H† T

(24)

is obeyed. If H is quasi-Hermitian, then the functional ε, ε (ψ) =

hψ|T H|ψi hψ|T |ψi

(25)

is greater than or equal to the lowest eigenvalue, ǫ0 , of H (H). Define the operator T as †

T = eω eω .

(26)

T = 1 + ω † (1 + ω) ,

(27)

Thus,

and it can be represented by a matrix of the structure LL† with a lower triangular matrix L whose diagonal elements are equal to one. Matrices of this type are positive definite (see for instance ref 29). Therefore, the operator T , being represented by a positive definite matrix, is positive definite. In addition, T and H obey Eq. (24). Therefore, H is quasi-Hermitian. Now let us choose |Ψi = P |φi

(28)

as a trial state. The variational functional becomes ε (φ) =

hφ| P T HP |φi . hφ|P T P |φi

(29)

Variation of the bra and a subsequent rearrangment exploiting the invertibility of P T P lead to the eigenvalue equation (P T P )−1 P T HP |φi = εP |φi, 10


(30)

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which can be written as Ω† Ω where

−1

Ω† HΩP |φi = εP |φi,

(31)

Ω=P +ω

(32)

has been introduced. With simple rearrangment Eq.

(31) can be transformed into an

eigenvalue equation for a Hermitian operator: Ω† Ω with |χi = Ω† Ω

1/2

|φi.

−1/2

Ω† HΩ Ω† Ω

−1/2

P |χi = εP |χi,

(33)

Eq. (31) and Eq. (33) are formally identical with Eq. (42) and Eq. (41) of ref 24, respectively. The latter are obtained by utilizing an ω dependent unitary transformation and thus, they are variational for all eigenstates. Therefore, it follows, that in our case quasiHermiticity implies variational property not only for the ground state but for all eigenstates. When solving the Schrödinger equations Eq. (31) and Eq. (33) by the basis expansion and truncation method the approximating eigenvalues obtained are variational with respect to both the basis size and the choice of ω as long as all matrix elements are calculated exactly. Thus, DVR variational with respect to both basis truncation and selection of grid points can be introduced with a particular choice of the basis set as described below. Insert the resolution of the identity operator, Eq. (6), between the operators T and H in Eq. (30) to obtain

P HP + (P T P )−1 P T QHP |φi = εP |φi.

(34)

Then, by splitting the Hamiltonian as in Eq. (2), and since due to ω satsifying the decoupling equation QsP = 0 the product approximation is exact, one can derive from Eq. (34) that

where

P H0 Ω + V (P sP ) + R |φi = εP |φi, R = Ω† Ω

−1

P T Q e−ω H0 eω P. 11


(35)

(36)

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For a moment let us assume that P H0 Q = 0. Then, the equation above reads as

P H0 P + V (P sP ) + R |φi = εP |φi.

(37)

By calculating the matrix representation of Eq. (37) in the biorthonormal basis {P |q¯α i, P |qα i}, with all matrix elements calculated exactly, the matrix eigenvalue equation X α

hq¯β |P H0 P |qα i + V (qα ) δβα + Rβα cα = εcβ ,

(38)

where Rβα = hq¯β |P RP |qα i,

(39)

cβ = hq¯β |φi,

(40)

and

results. By way of its construction this representation is variational with respect to both the basis size and the selection of grid points. If the terms Rβα are omitted, Eq. (38) reduces to the expression for the generalized DVR21,22,24 . The price paid for the omission of R is the loss of the variational property. Indeed, note, that the generalized DVR is, in fact, a representation of the non-Hermitian Hamiltonian operator P H0 P + V (P sP ). Since the variational principle30,31 does not apply to non-Hermitian Hamiltonian operators the generalized DVR should be nonvariational, but that can only happen if R does not vanish. We shall show that R can vanish under no circumstances. Since P T Q = ω † 6= 0, R cannot vansih unless Q (e−ω H0 eω ) P = 0, that is if ω is a solution of the decoupling equation for H0 . But ω has been chosen to satisfy the decoupling equation for the coordinate operators q. Since H0 and q do not commute Qe−ω H0 eω P 6= 0, R 6= 0 follows. As a result the generalized DVR (and, as a matter of fact, the standard DVR) is variational, neither with respect to truncation of the Hilbert space nor with respect to the selection of grid points. The generalized DVR is nonvariational approximation to the variational generalized DVR of Eq. (38). With R kept, Eq. (35) can be simplified as

Ω† Ω

−1

Ω† H0 Ω + V (P sP ) P |φi = εP |φi, 12


(41)

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whose matrix representation in the biorthonormal basis {P |q¯α i, P |qα i} gives the matrix eigenvalue equation X α

hqβ |H0 |qα i + V (qα ) δβα cα = εcβ ,

(42)

which is just the generalization of the variational DVR, Eq. (73) of ref 24, to a Hilbert space of infinite dimension. If we do not assume P H0 Q = 0 a new generalized FBR(DVR) is obtained from Eq. (35) by neglecting the operator R and calculating its representation in the biorthonormal basis {P |q¯α i, P |qα i}: X α

hq¯β |P H0 |qα i + V (qα ) δβα cα = εcβ .

(43)

Arguments similar to those given above show that this new generalized DVR is again nonvariational. It is a nonvariational approximation to the variational generalized DVR of Eq. (42). In passing it should be pointed out that although Eq. (42) has been useful for theoretical consideration, it is not suitable to numerical work for it contains Dirac’s delta functions. In any case, the problem of finding a variational generalized DVR suitable to numerical computations has been reduced to that of finding variational approximations to Eq. (42).

V.

SUMMARY While the generalized DVR allows one to employ different sets of grid points, the different

sets lead to results of different accuracies. The search for the optimal grid, the grid giving the possible highest accuracy results (eigenvalues), would be simpler if the DVR were variational with respect to the grid. Unfortunately, the generalized DVR (as well as the standard DVR) lacks this variational property. The reason for the generalized DVR failing to have variational property has been uncovered here by means of analytical derivation. The results presented have led to a new nonvariational generalized DVR (Eq. (43)) and the identification of the problem of finding a variational generalized DVR (that is a DVR which is variational over a large set of grid points and which is variational with respect to truncation of the Hilbert space) suitable to numerical work as a one of finding variational approximations to the variational generalized DVR given in Eq. (42). The generalized DVR of ref 21 and the new 13


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generalized DVR of Eq. (43) have been shown to be nonvariational approximations to Eq. (42).

ACKNOWLEDGMENTS The work described received support from the Hungarian National Office for Research and Technology under the contract ERC HU 09 OPTOMECH.

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