Laurence E. Fried and Gregory S. Ezra- PERTRUB: A Program for Calculating Vibrational Energies by...

8/3/2019 Laurence E. Fried and Gregory S. Ezra- PERTRUB: A Program for Calculating Vibrational Energies by Generalized Alg

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Computer Physics Communications 51 (1988) 103114 103

North-Holland, Amsterdam

PERTURB: A PROGRAM FOR CALCULATING VIBRATIONAL ENERGIES

BY GENERALIZED ALGEBRAIC QUANTIZATION

Laurence E. FRIED 1 and Gregory S. EZRA2Department ofChemistry, Baker Laboratory, Cornell University, Ithaca, NY14853, USA

Received 6 October 1987

We describe PERTURB, a special purpose algebraic manipulation program which calculates vibrational eigenvalues in

coupled oscillator systems. PERTURB implements the method ofgeneralized algebraic quantization (AQ), in which Van

Vieckperturbation theory is formulated in a mock phase space. The phase space formulation enables quantum and classical

perturbation theory to be treated on the same footing, and allows the systematic calculation of corrections to classical

perturbation results in powers of h. Generalized AQ is a powerful and efficient technique for calculating semiclassical

vibrational energy levels. In many cases, including just the ffrst correction to classical perturbation theory yields highly

accurate energies.

PROGRAM SUMMARY

Title ofprogram: PERTURB No. ofbits in a word: parameter in program. Set to 8 initially

Catalogue number: ABDN Numberoflines in combined program and test deck: 17298

Program obtainable from: CPC Program Library, Queens Uni- Separate documentation available: PERTURB users guide

versity of Belfast, N. Ireland (see application form in this

issue) Keywords: molecular vibrations, generalized algebraic quanti-

zation, semiclassical quantization, perturbation theory

Computerforwhich the program is designed and others on whichit is operable: PERTURB has been run on an IBM 3090, and Nature ofphysicalproblem

IBM 4381 a PRIME 9955, a Gould 9050 and an IBM PC. it PERTURB calculates energy eigenvalues for a set ofcoupled

should run with little or no modification on other systems harmonic oscillators.

MethodofsolutionComputer: IBM 3090; Installation: Cornell National Super- . . 0

computer Facility Generalized algebraic quantization, a perturbative techniqueallowing expansion in h.

Programming language used: C Restriction on thecomplexity oftheproblem

There is no pre-set limit on the size of the problem, although

High speed storage required: 350 Kb ofprogram storage, data the computer time required increases rapidly with the numberstorage varies with problem ofdegrees offreedom.

Mathematical Sciences Institute Fellow.2 Alfred P. Sloan Fellow.

OO1O-4655/88/$03.50 Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)


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104 L.E. Fried, G. S. Ezra /Programforcalculating vibrational energies

LONG WRITE-UP

1. Introduction perturbation theory (the famous small denomina-

tor problem [7]).Many semiclassical methods have been devel- Several approaches can be taken to avoid prob-

oped in recent years for calculating molecular lems associated with resonances. If the zerothvibrational and rotationvibration energy levels order system is close to resonance, but not exactly

(for reviews, see refs [15]).The present paper is resonant, the perturbation expansion will consist

concerned with perturbative methods [628],and of finite, but divergent, terms. Resummation

the extension of semiclassical perturbative quanti- [3638]is then a possibility [16,39]. From the

zation to arbitrary order in h [29]. In the semi- viewpoint ofcomplex analysis, the Taylor series in

classical perturbative approach, a series of canoni- perturbation parameter generated by classical per-

cal transformations is used to find good action turbation theory has a radius of convergence given

variables, to the desired order in perturbation. by the distance to the singularity (pole, branch

EBK quantization [1,3032]is then carried out by point) in the complex plane nearest to the origin.

assigning quantizing values to the good action Information about the singularity structure, if

variables [7]. More elaborate procedures are neces- available, can be incorporated into an approxi-

sary for resonant [11,13,1719,25]or nearly reso- mant that returns the original Taylor series when

nant [16,25,29] systems, as discussed below. The expanded about the origin. Pad approximationperturbative approach has several appealing fea- [40] is a well-known example; the Pad approxi-

tures. The results obtained are global; the energy mant replaces the Taylor series in perturbation

of any state may be found by substituting the parameter with a rational function, allowing polesappropriate quantum numbers into the simplified in the complex plane to be reproduced. If theHamiltonian. Vibrational eigenvalues obtained by divergence of the Taylor series is due to simple

this technique are accurate, even for excited states poles, Pad approximants can be expected to ac-

of realistic molecular potentials [25,29]. Moreover, curately resum the original perturbation expansion

the method has been found to give good results in [41].

the classically chaotic regime [11,25,28,29].This is A convergent resummation scheme may not,at first sight surprising, because EBK quantization however, be possible in the chaotic regime, since

is not expected to work in the chaotic regime, the good action variables sought via the perturba-

where good action variables and associated in- tion expansion do not exist globally [7]. Very littlevariant tori no longer exist [7]. The success of the is known about the analytic structure in the corn-

EBK quantization prescriptions in the chaotic reg- plex plane ofperturbation theory applied to multi-ime has been rationalized in terms of the existence dimensional systems.

of vague tori [11,28,34] or, more recently, canton Algebraic quantization (AQ) is an alternative toand broken separatrices [35]. resummation [1719,25].AQ avoids the small de-

Although the perturbative semiclassical quanti- nominator problem by allowing a more general

zation of realistic molecular systems is quite prom- form of the simplified Hamiltonian: terms de-

ising, several practical difficulties arise in its im- pending on up to N 1 [42] linearly independentplementation. The main problem is the ap- integer combinations of angles can be included.

pearance of resonances [7]. A resonance occurs The angle-dependent transformed Hamiltoman iswhen there is a commensurability between the then converted to a quantum operator. In general,fundamental frequencies of the system [7]. Low cross terms between coordinates and momenta

order resonances may lead to classical trajectories occur, so that a quantization rule [4347]is re-being greatly distorted from those of the zeroth quired to resolve the proper ordering of noncom-

order (uncoupled) system, even for small values of muting terms. For most quantization rules, the

the perturbation parameter. Resonances are also resulting Hamiltonian operator is block diagonal,responsible for the ultimate divergence ofclassical with finite blocks. A small matrix diagonalization


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LE. FriedG.S. Ezra /Programforcalculating vibrational energies 105

then yields energy cigenvalues. The AQ scheme 2. Algebraic quantization and the calculation ofhas several advantages. It extends the applicability vibrational energy levels

of perturbative semiclassical quantization to sys-

tems with up to N-i linearly independent nearresonances. Moreover, it is a uniform technique In this section we briefly review AQ and its

[48]; it reproduces inherently quantum splittings generalization to arbitrary order in h. Ordinaryquite accurately. Evidently, diagonalization of a AQ uses a series of canonical transformations tosmall matrix is sufficient to reproduce such level make the original Hamiltonian as nearly integra-

splittings. ble as possible. PERTURB carries out these trans-The operator corresponding to the transformed formations using the Lie transform algorithm of

Hamiltonian in the AQ approach clearly depends Dragt and Finn [52,53], described below. Reso-

in general upon the particular choice ofquantiza- nant and nearly resonant angle-dependent termstion rule, leading to noticeable disagreement be- are included in the transformed Hamiltonian. The

tween results obtained with different rules. We new Hamiltonian is converted to a quantum oper-

have shown how to incorporate corrections to ator, and a small matrix diagonalization is carried

arbitrary order in h into AQ [29], using a phase out to compute energy eigenvalues.

space formulation of quantum perturbation theory The schemejust described closely parallels con-

[46,47,49]. For polynomial Hamiltonians, the series ventional quantum mechanical Van Vleck per-

of corrections in /1 truncates, and results entirely turbation theory [50]. This close relationshipequivalent to quantum Van Vleck perturbation motivates the introduction of generalized AQ,

theory [50] are obtained. It is thereby possible to which is a formulation of quantum mechanical

explore the relation between quantum and classi- Van Vleck perturbation theory in a mock phase

cal perturbation theory in a systematic fashion. space [46,47]. The advantage of a phase spaceThe relative computational requirements of classi- formalism is that quantum and classical theories

cal versus quantum perturbation theory can also can be treated on the same footing.

be assessed. A mock phase space is induced by an invertibleA brief review of algabraic quantization and quantization rule [47]. The inverse quantization

the generalization to arbitrary order in h is given rule maps a quantum operator onto a unique /iin section 2. A general discussion of the structure dependent function on the mock phase space. ForofPERTURB, a program for implementing gener- example, a quantum mechanical commutator is

alized AQ for polynomial Hamiltonians of arbi- mapped onto an operator on the mock phasetrary dimensionality, is then given in section 3. space. If the Weyl quantization rule is used, this

Conclusions are presented in section 4. We note operator is called a Moyal bracket [33]. To lowestthere that a detailed discussion of generalized AQ order in h, the Moyal bracket is the Poisson

and extensive results are given in ref. [29]. The bracket. Thus, classical mechanics emerges natu-program described in the present paper is an rally in the mock phase space formalism. Gener-

extended version of the C program discussed in alized AQ uses Moyal brackets, rather than theref [Si]. The latter program only implemented Poisson brackets used in ordinary AQ, to carry

classical perturbation theory, whereas the current out coordinate transformations. The resulting the-

version of PERTURB is capable of calculating h ory is fully quanta!, and all transformations are

dependent corrections to the classical normal form. unitary. By truncating the Moyal bracket at a

For polynomial Hamiltomans, the series of correc- given order in /1, we arrive at approximations to

tions terminates, and results identical to quantum fully quantum perturbation theory.

Van Vleckperturbation theory can be obtained.


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106 L.E. Frie4 G. S. Ezra /Programfor calculating vibrational energies

2.1. Algebraic quantizarion sion. The transformed Hamiltonian resulting from

the elimination of all angle-dependent terms ex-

In AQ, a classical Hamiltonian is subjected to a cept those which are exactly resonant is called the

sequence of canonical transformations before it is BirkhoffGustavson normal form [59,60]. The

quantized. There are many ways to generate the presence of angle-dependent terms or divergent

transformations. We discuss briefly the method of corrections prevents perturbative EBK quantiza-Lie transforms [5458]as used in the algorithm of tion from being applied straightforwardly to reso-Dragt and Finn [52,53]. This technique is more nant or nearly resonant systems.

efficient than traditional techniques based on In AQ, angle-dependent terms correspondingmixed-coordinate generating functions [6,7], and is to exact and near resonances are left in the final

the algorithm used in PERTURB. Hamiltonian. If more than one linearly indepen-Lie transforms express a canonical transforma- dent resonant term is included in the final Ham-

tion entirely in terms of Lie operators. A Lie iltoman, the the classical problem is in generaloperator, L

1, is defined by nonintegrable [7]. As long as the number of reso-nant terms included Q is less than the number of

Lf_ (1 }, (1) degrees of freedom N, the problem can be treated

by AQ. AQ exploits the existence of N Qwhere ( , } denotes the Poisson bracket. Let K

ignorable angle vanables in the transformed Ham-be the Hamiltoman expressed in the transformed . . .iltoman. Each ignorable coordinate imphes thecoordinate system. Then . . . .

existence of a classical invanant which is a linear

K= exp(Lf)H, (2) function of action. A quantization rule (such asthe Weyl rule) is then used to associate a quantum

where H is the Hamiltonian in the original coordi- operator KQ with the transformed classical Ham-

nate system, and f is called a Lie generating iltoman K. Under mild assumptions about thefunction. Dragt and Finn use a sequence of such quantization rule, it can be shown that KQ also

transformations to simplify the original Hamilto- has N Q invanants which are linear functions ofthan: number operators ~[25]. The existence of these

quantum invariants implies that the operator KQ

K(z) H(i(z)) = exp( ~kj~) exp( e ~ is block diagonal [25]. The blocks are often small

x exp(eF )H(z) (3) and can easily be diagonalized to yield approxi-

mate quantum eigenvalues of the original system.where Fk is the Lie operator L1. The scheme outlined above has been applied to

The sequence of transformations is chosen to a variety of resonant, nearly resonant and nonres-

make K as close to integrable as possible. If an onant systems [25]. The 1: 1 HnonHeiles system

integer linear combination of zeroth order fre- has been quantized and quite good agreement

quencies m a vanishes identically, the original with variational quantum results obtained; purelyHamiltonian is resonant. The transformations (3) quanta! splittings were reproduced well [25]. Sys-

cannot convert a resonant Hamiltonian into a tems with more than two degrees of freedom have

function of good actions only. also been studied. For instance, the family of 3, 4

For nonresonant systems, all the angle depen- and 5 degree of freedom systems first studied by

dence of the transformed Hamiltoman Kcan by Skodje, Borondo and Reinhardt [61] has been

definition be eliminated to a given finite order quantized. Eigenvalues produced by AQ werewithout encountering divergence. Energy levels can found to be in good agreement with quantum

be found by substituting quantizing values of the variational calculations and/or other semiclassicalgood actions into K. For nearly resonant systems, methods.the angle dependence of H can be formally As mentioned above, the Haniiltonian operator

eliminated, but at the cost of introducing rapidly to be diagonalized in AQ depends on the quanti-

divergent corrections into the perturbation expan- zation rule used. This dependence on quantization


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L.E. Frie4 G.S. Ezra /Programforcalculating vibrational energies 107

rule is important if highly accurate vibrational As higher order corrections in /1 are included, the

eigenvalues are required [25,27]. For all of the dependence on quantization rule is decreased.systems mentioned above, it is found that an Generalized AQ is discussed in the next subsec-

approximation to the Weyl rule, first proposed by tion.

Robnik [18], gives more accurate results than those

obtained with the exact Weyl rule. The reason for 2.2. Generalizedalgebraic quantizationthis behavior is not understood, although it maybe significant that Robniks rule reduces to EBK The method of generalized AQ allows results

quantization for diagonal Harniltonians. Robniks from AQ to be successively improved until con-

rule breaks point group symmetries, however. A vergence to quantum mechanical Van Vleck per-hybrid of Robniks rule and the Weyl rule has turbation theory is achieved. A smooth transition

therefore been devised, which respects point group can then be made between the semiclassical and

symmetries while giving accuracy comparable to quantum limits. The incorporation of certain

that of Robniks rule. quantum effects (those analytic in It arising from

It is apparent that the dependence ofthe results near-identity transformations) is useful in solvingof AQ on quantization rule is undesirable. The vibrational problems to high accuracy [29]. It is

method of generalized AQ enables this depen- essential to note, however, that the entire problem

dence to be reduced in a systematic fashion. In is not subject to an expansion in It. Nonanalytic

generalized AQ, corrections which are of higher behavior in It can be introduced by the small

order in I t can be included in the normal operator. matrix diagonalization, so that generalized AQ

Van Vieck Generalized AQAQ

Quantum Symbol ClassicalOperator Function

Inverse RuleInput ~]Hamiltonian Classical Classical

Hamiltonian Symbol ~Hamiltonian4 Limit

Quint. Rule

Commutator Phase Space PoissonCommutator Bracket

Normal ~ Normal Approximate NormalOperator ____________ Symbol 4 Form

Small Matrix

Diagonalizalion

DiagonalHamilton ian

Fig. 1. Relation between Van Vieck perturbation theory, its mock phase space version and AQ.


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108 L.E. Fried, G.S. Ezra /Programforcalculating vibrational energies

avoids the convergence problems usually associ- A mock phase space version of Van Vieck

ated with expansion in It . perturbation theory is illustrated, in the center

Generalized AQ was formulated by analogy column of fig. 1. In the phase space approach, the

with quantum Van Vleckperturbation theory. The initial Hamiltonian operator is transcribed into a

relation between Van Vleck perturbation theory, function of 2N variables, called a symbol [62].

its mock phase space version and AQ is sum- The rule for transcription is invertible, so thatmarized in fig. 1. Van Vieckperturbation theory is every quantum operator is associated with a unique

conventionally formulated in terms of quantum symbol. While the use of quantization rules inoperators (left column of fig. 1). In the Van Vieck transcribing functions of p and q into operators is

approach, an initial Hamiltonian is simplified by a quite familiar, the inverse operation described here

sequence ofunitary transformations. Each of these may be less well known. As an explicit example,transformations can be expressed in terms of the the Weyl quantization rule applied to a monomial

commutator with a particular generating operator. in p and q gives:

This can be seen by a rearrangement of the ~fl(mn) 1

familiar form of Van Vieck perturbation theory; w(pmqn) = ~

the relation between the new Hamiltoman K and 10 2

the old Hamiltonian H is usually expressed as: (7)

1 kI

)The inverse relation can also be derived theK= . . . exp(~ckJ~)exp(~e fk-I Weyl symbol W~of a polynomial is

x . . . exp(~Efi)Hexp(-~-efi) W(pm~)min(m.n) (ih)

1 n!m!pm q

x . exp(~~e~1fk_l)exp(-~-~fk)..., =~=o 2l! (n l)!(m 1)!

( 4 ) ( 8 )

which can be rearranged with the relation Thus, there exists a one-to-one mapping fromsymbol to quantum operator. This also serves to

exp(f) ii~exp( f) = exp( [f, ] ) H, (5) illustrate that finding a symbol is not the same astaking the classical limit, since the latter procedure

to give would map many operators onto the same classi-cal function.

j~. . . exp(~[J~ i ) exp(-~[J, ]) n . (6) The mapping between symbols and operatorscan be used to produce an alternative formulationof quantum mechanics, based entirely on symbols.

This sequence of transformations is the quantum To do this, the equivalent of operator multiplica-analogue ofDragtFinn perturbation theory, since tion must be found in the mock phase space. This

(1/i h )[ , ] corresponds to the Poisson bracket. multiplication is sometimes referred to as a twistedThe sequence of transformations is chosen to multiplication [62,63]. In the case of the Weyl rule,

render the final Hamiltoman as close to diagonal the twisted multiplication (denoted by *) of two

as possible. We term the resulting simplified Ham- symbols A~and B~is:

iltoman a normal operator. If generalized Fermi

resonances are encountered, the normal operator I i I t*B~= exp

will not be diagonal. Nonetheless, if sufficiently~Pe

~1

few resonances are found, the normal operator xA~(q~,pA)BS(qB, PB). (9)will have a block diagonal matrix. Therefore, only

small matrix diagonaiizations are necessary to A~*B~,when converted to an operator, is theproduce a diagonal Hamiltonian. same as the operator AB, so that twisted multipli-


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L.E. Frie4 G.S. Ezra /Programforcalculating vibrational energies 109

cation is entirely equivalent to operator multipli- perturbation theory presented here is entirelycation. In a similar manner, the commutator maps equivalent to the usual Hilbert space version. The

into a Moyal bracket, given by phase space version, however, lends itself to ex-A B A B B *A ~ pansion in It . In generalized AQ, the expression1 . S S IS = S * S S S for the Moyal bracket is truncated at a given order

In Van Vleckperturbation theory formulated in in It . As noted above, the full problem is nota mock phase space, the Hamiltonian symbol is expanded in It . Thus, the small matrix diagonali-

simplified by a sequence oftransformations gener- zation can introduce nonanalytic behavior in It ,

ated by phase space commutators, replacing the thereby allowing important effects such as tunnel-

Van Vleck generating operators with generating ing to be described.

symbols. By this sequence of transformations, a To lowest order in It , generalized AQ is the

simplified Hamiltonian symbol is arrived at, which same as ordinary AQ. This point is illustrated inwe term a normal symbol. A Hamiltonian symbol the rightmost column of fig. 1. AQ uses a se-

H5 will correspond to a diagonal operator H if quence of canonical transformations to simplify a

and only if J-I~is a function of the products a7a, classical Hamiltoman. The limit as It 0 of the

only, where a and a * are the classical analogues Hamiltoman symbol isjust the classical Hamilto-

of creationamuhilation operators. than, so that the starting point of AQ is arrived at

We seek a sequence ofunitary transformations by taking the classical limit of fully quantumwhich renders the Hamiltonian as nearly diagonal phase space perturbation theory. It is not always

as possible. This is expressed in mock phase space necessary to take this limit; if no momentumco-

as ordinate cross terms are present in the quantum11 Hamiltonian, the initial Hamiltoman symbol will

K5 = . . . exp~~~~Ek(fk,}s) be the same as the classical Hamiltonian.

1 The classical limit of the phase space commuta-xexp(-_E~(fk_i~.}~) tor is the Poisson bracket. Unitary transforma-

iIt lions of the quantum theory therefore map into

x exp( - 4 - ~ } H~. (11) canonical transformations. The simplified classical\iIt S1 Hamiltonian is termed a normalform [59]. It can

The procedure for choosing the generating func- be viewed as an approximation to the normal

tion fk is similar to that of the classical theory. symbol. Hence, a quantization rule can be used to

On the kth iteration, we choose fk to make the derive an approximate normal operator corre-

transformed Hamiltonian a function of a.a7. Res- sponding to the normal form.

onances between the zeroth order frequencies lead Several multimode systems have been treated

to complications, just as they do classically, with generalized AQ [29]. We have found a sub-

Explicit formulas can be derived by expanding stantial increase in the accuracy of low-lying ei-

all quantities in powers of e. Let H be the genvalues when the first correction in It is in-Hamiltoman symbol after k transformations. Ex- cluded. The most striking example is a 3 modepanding H in powers of e yields model for rotationless 03. Ordinary AQ shows a

10 cmt error in the ground state energy of 0,

rrk_V j k I ~ii ,~E r1,. t,1B) when the Weyl quantization rule is used. In-

Inserting this into eq. (11) gives cluding just the first correction in It, however,reduces this error to 0.02 cm [29,641.Evidently,

int(j/k) / . . .

H ~- ~ }sHk_I 113\ higher order terms in It are needed to obtain high i~o (iIt)ll Jlk 1 accuracy, especially for small quantum numbers.

Furthermore, for many systems, the first correc-Once the normal symbol is found by eq. (13), it tion is the most important. In the following sec-

is mapped into a normal operator by the quantization, we will discuss how generalized AQ is imple-

tion rule. The phase space version of Van Vieck mented in the program PERTURB.


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110 L.E. Frie4 G.S. Ezra /Programfor calculating vibrational energies

3. PERTURB by complex exponentials of angle variables. Thus,every term is of the form

PERTURB is a special purpose program writ-

ten in C for carrying out generalized AQ. PER- (a+ ib)joao . . . j~Nexp(i(90n0+

TURB is compatible with any Ccompiler support- (14)

ing the proposed ANSI standard. Alternatively,the Kernighan and Ritchie standard [65] may be The symbols used in generalized AQ can also be

used, with the extensions of structure assignment expressed in this form. Terms of the above formand structure-valued functions. UNIX C is an can be quickly transcribed into the classical ana-

example of a non-ANSI C compiler under which logues of creationannihilation variables, by the

PERTURB will run. The current version of the prescriptionprogram finds vibrational eigenvalues, normal

symbols and generating symbols. Future versions a, = (~~~/2exp( i9,). (is)will also find wavefunctions and properties such

as dipole matrix elements. The program is mod- PERTURB represents a term by a structure,

ular, with a language-like syntax being employed called struct store. Structures group data into a

in the symbolic manipulator. The program has single object. Struct store contains the coefficient

many o p t i o n s . U s e r s c a n c h o o s e r e p r e s e n t a t i o n s o f (a +ib) as a complex data type, the integer vectora n e x p r e s s i o n ( s e e b e l o w ) w h i c h s a v e memory a t 2 a , a n d t h e i n t e g e r v e c t o r n. The u s e o f s t r u c t u r e s

the expense of CPU time, or use a representation is an important advantage in programming in C.

which allows maximum speed. The initial Ham- Once a set of functions that handle the structures

iltoman is entered either as a polynomial in p and has been constructed, the structure can be regarded

or directly in terms of zeroth order action-angle as a user-defined data type in its own right. The

variables. The Hamiltonian is then subjected to a details of manipulating the components of the

sequence of canonical/unitary transformations. structure can then be forgotten. This philosophy

After the transformations have been carried out, of programming is essential to the design of PER-

a normal symbol is written to file. If desired, TURB.PERTURB will then calculate eigenvalues by con- PERTURB handles expressions that are sums

structing blocks, converting the normal symbol to of terms. An expression could therefore be repre-

an operator, and diagonalizing t h e o p e r a t o r w i t h i n s e n t e d b y a n a r r a y ofterms. In C, it is possible to

t h e b l o ck s , allocate this array dynamically, s o t h a t n o mem-The program consists of a special purpose alge- ory is wasted. Dynamic memory allocation is a

braic manipulator and accompanying numerical substantial advantage of C over Fortran. It isroutines for block diagonalization. In this section convenient, however, to give an expression ad-

we discuss the design of the program, and the ditional characteristics. For instance, PERTURB

algorithms used in carrying out calculations. More can treat expressions as being of either temporary

detailed information, such as input files and sam- or permanent interest. Temporary expressions areple output, is available in a users guide accompa- automatically deallocated (the memory associated

nying the program [66]. with them is freed for other use) as soon as theyare used in an expression-producing function. Per-

3.1. The algebraic manipulator manent expressions, however, are not automati-cally deallocated. The use of temporary expres-

The core ofPERTURB isan algebraic mampu- sions allows PERTURB functions to be conveni-lator. Although PERTURB can accept polynomi- ently chained, as will be discussed below.

als in Cartesian variables as input, it converts Thus, the structure representing an expression,

them into an internal form which is convenient for called struct express in PERTURB, contains aclassical perturbation theory. This form consists of pointer to an array of terms (a pointer is a varia-

half-integer powers of action variables multiplied ble that contains another variables machine ad-


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LE. Frie4 G.S. Ezra /Programfor calculating vibrational energies 1 1 1

dress), the number of terms in the expression, and balancing [67]. The AVL algorithm requires that

its storage classification. As a convenience, the the depths of the offspring nodes differ by at mostfinal representation of an expression is a pointer one. If this is not the case after an insertion has

to struct express. This allows a PERTURB func- been made, the nodes ofthe tree are rearranged bytion to modify the contents of the structure associ- balancing functions.

ated with an expression. This pointer is said to There are several ways to customize PERTURBhave type EXPR in PERTURB. to fit the requirements of a particular problem.

Functions in PERTURB do basic operations One of the most important parameters is RFLAG.such as addition, multiplication and differentia- When RFLAG is TRUE, PERTURB assumes that

lion. In addition, there are special purpose routines all expressions are real. This allows a term to

for computing the Poisson bracket and the Moyal represent both itself and its complex conjugate, so

bracket. These functions have a modular, Ian- that the size of every expression is reduced byguage-like design. PERTURB manipulation func- approximately a factor of two.

tions take arguments of type EXPR, and return In addition, the size of a term can be modified

values of type EXPR. This allows functions to be to make more effective use of memory. This is an

conveniently chained. The following code frag- important consideration when doing high order

ment adds two EXPRs c and d, multiplies them perturbation theory. A term is declared to be big

by a, and puts the result in c: c = mult(add(c, d), a). enough to handle problem with dimensionality

The storage classification of the return value of a MAXNDOF. Therefore, memory use can be opti-

function is always temporary. Therefore, the inter- mized by making the maximum number ofdegreesmediate result add(c, d) is automatically deallocat- of freedom, MAXNDOF, equal to the dimen-

ed by multO. sionality of the problem at hand. There are otherThe PERTURB manipulator is very efficient, ways to optimize the use of memory. One is to

We have obtained part of this efficiency by using change the parameter FIELDTYPE. The small

a sophisticated sorting algorithm. As each succes- integers representing the powers and exponents ofsive term in an expression is calculated, it is a term can be stored in either a short integer (2

placed in a binary tree structure. A binary tree bytes), or character (1 byte) by making FIELD-

consists of a root, where a term is stored, and two TYPE equal to either SHORTFIELD or CHAR-offspring binary trees, The offspring trees may be FIELD. CHARFIELD saves on memory, but at a

empty. Before a term is placed into a tree, an small cost in efficiency on most systems. The

array of sorting keys is calculated. Then, when the coefficients in a term can also be optimized forterm is inserted into the tree, the sorting keys are memory use, IfRFLAG is TRUE, setting COEF-

compared to the keys of the root. If the first FTtPE to NEWCOEFF will give a more corn-nonmatching key of the term is greater than that pact, but somewhat slower, form for the coeffi-

of the root, the term is inserted into the right cients.

offspring. Otherwise, it is inserted into the left PERTURB also allows the user to choose the

offspring. This process of comparison continues, way in which resonances are handled, In solvinguntil a term is either placed into an empty node, for the normal form/symbol, PERTURB must

or matched with an existent term (in which case decide whether a given term is resonant or not.

the coefficients are added). There are two ways of doing this, called the ab-Primitive insertion into a binary tree can be solute criterion and the convergence criterion, re-

very efficient, as long as the terms arrive in a spectively. Under the absolute criterion, a term is

random order. If the terms arrive in order, how- considered resonant if the absolute value of the

ever, the depth of the tree grows linearly with the denominator is ~ is less than a given amount,number of terms inserted, rather than logarithmi- where is is the angle vector appearing in the term,

cally. This results in extremely slow sorting. PER- and w is a vector containing the zeroth orderTURB avoids this situation by using a nonrecur- frequencies. This criterion is simple, but it does

sive implementation of theAVL algorithm for tree not take into account the size of the term,


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112 L.E. Frie4 G. S. Ezra /Programfor calculating vibrational energies

Under the convergence criterion, on the nth found in the normal form into a matrix. The

transformation, a term is considered resonant if coefficients of the invariants are then given by thenull vectors of this matrix. Once the invariants

i i C I I( I > -~, (16) have been found, they are simplified (i.e., reduced(n I I to a form with integer coefficients) by a trial and

error algorithm, which works well in systems withThis criterion attempts to force the calculation to a few degrees of freedom. If the system has moreconverge geometrically with a rate 1/y. A very than 5 degrees of freedom, however, the timesmall value of ~ will produce a rapidly convergent required to simplify the invariants is prohibitive.

perturbation expansion. It is also likely, however, PERTURB allows the user to specify whether the

that for small values of y so many linearly inde- invariants and frequency vectors should be sim-

pendent resonant terms will arise that the final plified or not.

Hamiltonian will have no invariants at all. On the To calculate eigenvalues, the user specifies an

other hand, if y is chosen to be too large, the energy range of interest, and PERTURB thenperturbation expansion will converge too slowly to finds all states with zeroth order energies in this

yield useful information on excited states. (Of range. The invariants associated with each state

course, if y is chosen to be too large, the perturba- are evaluated, and the list of states is sortedtion expansion could diverge.) As a general rule, according to the invariants, so that states in the

we find that y should be roughly the size of the same block are adjacent to one another in the listperturbation parameter (assuming dimensionless of states.coordinates and a zeroth order Hamiltonian of The list of states is then passed to a diagonali-

order unity). This represents the convergence rate zation routine, which evaluates the Hamiltonianone would naively expect if small denominator matrix for each block of states. The matrix ele-

problems were not present. We find that the con- ments can be either evaluated straightforwardly,

vergence criterion usually leads to better results or resummed by the epsilon algorithm [40]. The

than the absolute criterion [29]. user indicates which quantization rule should beThe user can limit the number of resonant used in evaluating the matrix. There are three

terms included in the normal form, Normally, up choices: the Weyl quantization rule [43], Robniksto N 1 linearly independent resonant terms are approximation to the Weyl rule [18] and the sym-

eliminated. This is the maximum number con- metry-preserving semiclassical (SPSC) quantiza-

sistent with a block diagonal Hamiltonian. tion rule [29]. As noted in the previous section,

Robniks approximation to the Weyl rule given3.2. The block diagonalization routines more accurate results than the Weyl rule itself for

many systems [29], while breaking point groupIn addition to the algebraic manipulator, PER- symmetries. The SPSC rule is a hybrid ofRobniks

TURB contains routines which find linear in- rule and the Weyl rule [25], which gives results of

variants and diagonalize the Hamiltonian within accuracy comparable to Robniks rule while re-

the blocks formed by the invariants. The block specting point group symmetries. If semiclassical

diagonalization can be used either with or without quantization ofan angle-independent normal form

an accompanying perturbative calculation. Thus, is desired, Robniks quantization rule should bepre-existing normal forms can be diagonalized chosen. In the current version of PERTURB, if

with PERTURB. The block diagonalizer can also the perturbative transformations have been car-be used to do an ordinary diagonalization of an ried out to higher order in It , the Weyl quantiza-

arbitrary polynomial Hamiltonian. tion rule must be used. A mock phase spaceWhen first passed a normal form, the block implementation of other quantization rules will be

diagonalizer determines the linear functions of added in the future.

action, if any, which commute with it. This is done Once the matrix associated with a block hasby placing all linearly independent angle vectors been evaluated, PERTURB diagonalizes it by the


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114 LE. Frie4 G.S. Ezra /Programfor calculating vibrational energies

[31] M.L. Brillouin, J. Phys. 7 (1926) 353. [51] L.E. Fried and G.S. Ezra, J. Comput. Chem. 8 (1987) 397.

[32] J.B. Keller, Ann. Phys. 4 (1958) 180. [52] A.J. Dragt and J.M. Finn, J. Math. Phys. 17 (1976) 2215.

[33] J.E. Moyal, Proc. Camb. Phil. Soc. 45 (1949) 99. [53] AJ. Dragt and J.M. Finn, J. Math. Phys. 20 (1979) 2649.

[34] W.P. Reinhardt, J. Phys. Chem. 86 (1982) 2158. [54] J.R. Cary, Phys. Rep. 79 (1981) 129.

[35] M.J. Davis, J. Phys. Chem. 92 (1988) 3124. [55] G. Hori, Publ. Astron. Soc. Japan 18 (1966) 287.

[36] C.M. Bender, Intern. J. Quantum Chem. 21(1982) 93. [56] A. Deprit, Celest. Mech. 1(1969)12.[37] T.T. Wu, Intern. J. Quantum Chem. 21(1982)105. [57] J. Henrard, Celest. Mech. 3 (1970) 107.

[38] B. Simon, Intern. J. Quantum Chem. 21 (1982) 3. [58] R.A. Howland Jr., Celest. Mech. 15 (1977) 327.[39] T. Banks and C.M. Bender, J. Math. Phys. 13 (1972) 1320. [59] G.D. Birkhoff, Dynamical Systems, vol. 9A (M.S. Col-

[40] G.A. Baker Jr., Essentials of Pad Approximants loqium Publications, New York, 1927).

(Academic Press, New York, 1975). [60] F.G. Gustavson, Astron. J. 71 (1966) 670.

[41] Pad approxiniants can also mimic branch cuts with a l ine [61] R.T. Skodje, F. Borondo and W.P. Reinhardt, J. Chem.

ofpoles. Phys. 82 (1985) 4611.

[42] N is the number ofdegrees offreedom. [62] A. Voros, Ann. Inst. Henri PoincarC 24 (1976) 3.

[43] H.Weyl, The Theory ofGroups and Quantum Mechanics [63] A. Voros, Ann. Inst. Henri Poincar 26 (1977) 343.

(Dover, New York, 1950). [64] The first order term in the expansion of the Moyal

[44] S.R. de Groot and L.G. Suttorp, Foundations of Elec- bracket corresponds to theclassical commutator; in order

trodynamics (North-Holland, Amsterdam, 1972). to obtain corrections to classical perturbation theory, it is

[45] R. Abraham and J.E. Marsden, Foundations of Mocha- necessary to include the thirdorder term in It(the second

nics (Benjamin/Cummings, Reading, 1978). order term is zero). This third order term is the first

[46] N.L. Balazs and B.K. Jennings, Phys. Rep. 104 (1984) 347. correction referred to here.

[471M. Hillery, R.F. OConnell, MO. Scully and E.P. Wigner, [65] R.W. Kernighan and D.M. Ritchie, The C ProgrammingPhys. Rep. 106 (1984) 121. Language (Prentice-Hall, Englewood Cliffs, 1978).[48] C.C. Martens and G.S. Ezra, in: Tunneling, eds. J. Jortner [66] The PERTURB Users Guide is in preparation.

and B. Pullman (Reidel, New York, 1986). [67] R.L. Kruse, Data Structures and Program Design (Pren-

[49] S.S. Mizrahi, Physica A 127 (1984) 241. tice-Hall, Englewood Cliffs, 1984).

[50] D. Papou~ekand M.R. Aliev, Molecular VibrationalRo- [681 W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T.tational Spectra (Elsevier/North-Holland, New York, Vetterling, Numerical Recipes (Cambridge Univ. Press,

1982). London, 1986).

Laurence E. Fried and Gregory S. Ezra- PERTRUB: A Program for Calculating Vibrational Energies by...

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