Generalized rotating wave function for quantum Monte Carlo

PHYSICAL REVIEW A, VOLUME 61, 012506

Generalized rotating wave function for quantum Monte Carlo calculations of rovibrational levels of n-body systems L. S. Costa,* Frederico V. Prudente,† and Paulo H. Acioli Nu´cleo de Fı´sica Atoˆmica, Molecular e de Fluidos, Instituto de Fı´sica–Universidade de Brası´lia, CP 04455, 70919-970 Brası´lia-DF, Brazil 共Received 1 June 1999; published 13 December 1999兲 We propose a generalization of the method proposed by Prudente and Acioli 关Chem. Phys. Lett. 302, 249 共1999兲兴 to compute rovibrational spectra of molecular systems. We present the expressions of the generalized rotating wave function for n-body systems. We discuss the advantages of the current approach over traditional variational methods. We also obtain an important relation for the Euler angles in terms of the Cartesian components of Jacobi vectors, expressions that are needed to compute the matrix elements utilizing the correlation function quantum Monte Carlo method. PACS number共s兲: 33.20.Wr

I. INTRODUCTION

The study of the rovibrational spectra of molecules and clusters involves the solution of hard to solve partial differential equations, either within classical or quantum mechanics. To obtain the solutions of this problem we must address two main issues: 共i兲 the excessive computational effort and 共ii兲 the search for convenient coordinate systems. Usually, quantum-mechanical calculations of excited rovibrational energy levels are performed using variational methods. Variational methods expand the wave function in an appropriate basis set, evaluate the required integrals using quadrature or pointwise methods, and solve the resulting eigenvalue problem in a subspace spanned by the basis set. In principle, an exact solution of the problem can be obtained. However, the number of basis functions and the computational integrating effort grow exponentially with the number of particles, thus making an accurate treatment of larger systems very difficult. For systems of more than three bodies, one of the main difficulties is to obtain an expression for the kinetic-energy operator in the chosen coordinates. Many years ago, Watson 关1,2兴 completed the solution of this problem in normal coordinates for any polyatomic molecule. However, a consensus has since emerged that internal coordinates must be used in variational calculations to bypass, for example, the unfactorizability of Watson’s kinetic-energy operator and the problem of choosing a reference geometry to define normal coordinates 关3,4兴. Many exact forms for the kinetic-energy operator in various coordinate systems 共e.g., valence, Jacobi, Radau, and hyperspherical兲 for four atoms have now been derived without approximation from the nonrelativistic Laplacian form in space-fixed Cartesian coordinates 关5–13兴. The description of an N-particle system is of great importance and a great challenge 关14–16兴, so a generalized manner to treat these systems is highly desirable.

*Corresponding author. Electronic address: [email protected] † Present address: Departamento de Quı´mica, Universidade de Coimbra, P-3049 Coimbra Codex, Portugal.

1050-2947/99/61共1兲/012506共7兲/$15.00

In the traditional methods used to treat systems of four atoms the exact quantum calculation has been shown to be extremely difficult. Only recently have calculations of rovibrational energy levels of tetraatomic molecules been reported 关7,17–19兴. In general, these methods look to solve the largest possible number of degrees of freedom analytically in order to minimize the computational effort. For this, it is necessary to write the Hamiltonian in such a manner as to contain explicitly these degrees of freedom. Usually, six degrees of freedom are solved analytically: three of them are related to the center-of-mass translational motion, and the other three are related to the spatial rotation of the system. The solution for the center-of-mass translational problem is straightforward, and the Euler angles are the natural choice to describe the rotational coordinates. In this way, the Hamiltonian contains the angular-momentum operator. We can use the Wigner D functions as solutions associated with the rotational motion and we can integrate these three degrees of freedom analytically. The analytical integration of the kinetic energy for other degrees of freedom is also available for particular choices of coordinate system 关20兴. However, for systems of more than four atoms this approach has been shown to be very difficult. In addition, the geometric visualization of the real system is unfeasible using these sophisticated coordinate systems of four or more atoms, thus making us look for more powerful methods for the calculation of multidimensional systems. An alternative approach is the use of stochastic methods to compute the multidimensional integrals that appear in the time-independent problems. With these methods the computational effort to make those integrals does not grow exponentially with the number of particles. In particular, the quantum Monte Carlo techniques are based on a stochastic principle 关21兴. These techniques have been employed with great success in the calculation of the properties of groundstate and finite-temperature quantum systems. However, there are a few studies of excited states. The use of the correlation function quantum Monte Carlo 共CFQMC兲 method, developed by Bernu and co-workers 关22,23兴 to calculate the excited vibrational states of triatomic 关23–25兴 and tetraatomic 关23,24兴 molecules, the study of multidimensional tunneling motion in complexes using the rigid-body diffusion Monte Carlo 共RBDMC兲 method 关26,27兴, and the use of the projection operator imaginary time spectral evolution

61 012506-1

©1999 The American Physical Society

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共POITSE兲 关28兴 to calculate the rovibrational states of molecular systems 关29兴 are some examples of excited-state studies. Another important characteristic of these methods is that we can use simple nonseparable analytical basis functions and that it is not necessary to reduce the number of degrees of freedom that will be integrated numerically. Thus, we can work in the Cartesian coordinate system where the geometric visualization of the real system is possible. An alternative procedure to calculate the excited rovibrational energy levels based in the CFQMC method was recently developed by Prudente and Acioli 关30兴. For this, they proposed the utilization of rotating functions as the trial basis functions in the CFQMC method and have studied two lowdimensional systems that showed the accuracy of method: the rotating harmonic oscillator and rotating Morse potential. In this work we generalized for N atoms the method proposed by Prudente and Acioli and defined the generalized rotating wave functions. In Sec. II, we summarize the CFQMC method. The rotating wave function for lowdimensional systems is reviewed in Sec. III. In Sec. IV we show the generalization to the n-body problem of this procedure. The important relation for Euler angles in function of Cartesian components of Jacobi vectors is obtained in Sec. V and the last section is dedicated to concluding remarks.

m

兺

␤ ⫽1

ជ 兲 ⫽E i ⌽ i 共 Rជ 兲 , H⌽ i 共 R

lim ⌳ k 共 t 兲 ⫽E k ,

ជ ) 其 has some overlap with the as long as the basis set 兵 f ␣ (R eigenstates 兵 ⌽ i 其 and is linearly independent. The matrix elements defined in Eq. 共3兲 are evaluated using Monte Carlo methods. For this, we generate random walks according to the diffusion equation in imaginary time 关31兴 using the following updating scheme of the coordinates at a time step ␶ : ជ i⫹1 ⫽Rជ i ⫹ ␶ d⌿ ⫺1 共 Rជ i 兲 ⵜ⌿ 共 Rជ i 兲 ⫹ 共 2 ␶ d 兲 1/2␹ i , R

H⫽⫺

兺

i⫽1

冋

冕 ជ 冕 ជ

N ␣␤ 共 t 兲 ⫽ H ␣␤ 共 t 兲 ⫽

ជ 2 H f ␣ 共 Rជ 2 兲 e dR 1 dR

n ␣␤ 共 k ␶ 兲 ⫽

respectively. Associated with these matrices, the generalized eigenvalue problem is defined as

ជ j 兲 ⫹E L 共 Rជ j⫹1 兲兴 , 关 E L ⌿共 R ⌿ 共7兲

1 2p

p

兺 关 F ␣共 Rជ i 兲 F ␤共 Rជ i⫹k 兲

i⫽1

ជ i⫹k 兲 F ␤ 共 Rជ i 兲兴 W i,i⫹k , ⫹F ␣ 共 R h ␣␤ 共 k ␶ 兲 ⫽

1 4p

p

兺 关 F ␣共 Rជ i 兲 F ␤共 Rជ i⫹k 兲 E L 共 Rជ i⫹k 兲 i⫽1 ␤

ជ i 兲 E L 共 Rជ i 兲 ⫹F ␣ 共 Rជ i⫹k 兲 F ␤ 共 R ␤ ជ i 兲 F ␣ 共 Rជ i 兲 F ␤ 共 Rជ i⫹k 兲 ⫹E L ␣ 共 R ជ i⫹k 兲 F ␣ 共 Rជ i⫹k 兲 F ␤ 共 Rជ i 兲兴 W i,i⫹k , ⫹E L ␣ 共 R

共8兲

where

共3兲

ជ 1兲, f ␤共 R

兺

j⫽n

where E L ⌿ ⫽⌿ ⫺1 H⌿ is the local energy function. The matrix elements, after symmetrization, are evaluated as

ជ 2 f ␣ 共 Rជ 2 兲 e ⫺tH f ␤ 共 Rជ 1 兲 , dR 1 dR ⫺tH

册

n⫹k⫺1

W n,n⫹k ⫽exp ⫺0.5␶

共2兲

ជ is the vector of 3(N⫺1) where ␮ i is the reduced mass, R ជ ) is the potential energy, and E i and ⌽ i are coordinates, V(R the eigenvalues and eigenvectors of H. In the CFQMC method used to solve Eq. 共1兲 we start with ជ ) 其 , defining the a trial basis set of m known functions 兵 f ␣ (R overlap and Hamiltonian matrix elements as

共6兲

where d⫽ប 2 /2␮ , ⌿ is a guiding wave function, and ␹ i is a normally distributed random variable vector with zero mean and unit variance. After successive applications of Eq. 共6兲 we obtain a ‘‘trajectory’’ in phase space with probability ជ ) 兩 2 . To determine the matrix elements, each fragment 兩 ⌿(R of the trajectory must be weighted by

共1兲

ប2 2 ជ 兲, ⵜ ⫹V 共 R 2␮i i

共5兲

1⭐k⭐m,

t→⬁

where the Hamiltonian associated with the relative motion of particles 共excluding the kinetic energy of the center of mass兲 关15兴 is given as N⫺1

共4兲

with d k (t) being the kth eigenvector and ⌳ k (t) its associated eigenvalue. It has been shown 关22,23兴 that

II. CORRELATION FUNCTION QUANTUM MONTE CARLO METHOD

Before introducing the rotating wave function, let us describe the numerical method utilized in this approach, the correlation function quantum Monte Carlo method. To obtain the rovibrational energy levels is necessary to solve the following eigenvalue problem:

关 H ␣␤ 共 t 兲 ⫺⌳ k 共 t 兲 N ␣␤ 共 t 兲兴 d k ␤ 共 t 兲 ⫽0,

ជ 兲 ⫽ f ␣ 共 Rជ 兲 /⌿ 共 Rជ 兲 , F ␣共 R

共9兲

ជ 兲 H f ␤ 共 Rជ 兲 . E L ␤ 共 R 兲 ⫽ f ␤⫺1 共 R

共10兲

As p→⬁, n ␣␤ (k ␶ ) and h ␣␤ (k ␶ ) converge to N ␣␤ (k ␶ ) and H ␣␤ (k ␶ ). After the system has reached the stationary state the generalized eigenvalue problem is solved and the rovibrational spectrum is obtained.

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PHYSICAL REVIEW A 61 012506

The variational Monte Carlo 共VMC兲 consists of considering t⫽0 in Eq. 共3兲. Thus, the trajectory weight W is always equal to 1 and in the matrix elements, Eq. 共8兲, the constant k is equal to 0. The variational eigenvalues and eigenfunctions are obtained solving the generalized eigenvalue problem, Eq. 共4兲, at t⫽0. III. ROTATING TRIAL FUNCTIONS

In Ref. 关23兴, Bernu et al. presented a simple way of generating the trial functions 兵 f ␣ 其 that are used to calculate the vibrational energy levels. With the method, the basis functions are directly expressed in terms of interatomic distances. Acioli and Soares Neto 关24兴 proposed a similar method, with a few differences in the excited-state trial functions. In Ref. 关30兴, the following procedure was proposed to generate rotating trial basis functions: 共i兲 Start from a simple approximation to the excited vibrational state eigenfunctions 兵˜f i 其 共for example, the Acioli et al. trial basis兲. 共ii兲 Using the VMC method, calculate a new orthonormal vibrational basis 兵 f i 其 . 共iii兲 The rotating trial basis set 兵 f ␣ 其 is obtained as a combination of an 兵 f i 其 function and the eigenfunctions of the Jˆ2 operator 共e.g., spherical harmonics for the spherical symmetrical potential and Wigner functions for general systems 关32兴兲. This procedure was applied to diatomic molecules, and for this kind of system the initial vibrational basis functions are given by 1 ˜f i 共 r 兲 ⫽ 共 r⫺r e 兲 i e ⫺ ␣ (r⫺r e ) 2 , r

i⫽0, . . . ,N.

共11兲

Utilizing the VMC method, the new orthonormal vibrational basis set is

A similar extension of the triatomic system is indicated in the same paper. In the following section we generalized this procedure to an n-body problem utilizing the mass weighted Jacobi coordinate system 关15兴. Although we chose this coordinate system, we must emphasize that this is an arbitrary choice, and the procedure can be realized in any set of vectors that determine the coordinate system for the N-particle problem. We also observe that our procedure is made in one clustering scheme ␭ of the mass weighted Jacobi coordinate system 关9兴, but it is invariant by kinematic rotations 关33兴. IV. GENERALIZED ROTATING WAVE FUNCTION

Let us consider a system of n bodies located in the space ជ c.m. ,rជ ⬘ ␭(1) , . . . ,rជ ⬘ ␭(N⫺1) , spanned by N Jacobi vectors R where Rជ c.m. is the position vector of the center of mass of the system with respect to the fixed origin in the space O, and rជ ⬘ ␭( j) 共with j⫽1, . . . ,N⫺1) is a set of vectors of relative position between the center of mass of the bodies and a clustering scheme ␭. Moreover, consider the following associated set of masses (M , ␮ ␭(1) , . . . , ␮ ␭(N⫺1) ) and define the following mass weighted Jacobi coordinate system: rជ ( j) ⫽

兺 j⫽0

c i j˜f j 共 r 兲 ,

i⫽0, . . . ,n v ib ⫺1.

␮ ␭( j) ␮

1/2

rជ ⬘ ( j) ,

共15兲

where M is the total mass of the system, and

冋兿 册 N⫺1

␮⫽

j⫽1

1/(N⫺1)

␮ ␭( j)

冋

N

1 ⫽ m M i⫽1 i

兿

册

1/(N⫺1)

共16兲

is an effective reduced mass of the system that is independent of the ␭ arrangement. In this coordinate system the kinetic-energy operator associated with the total relative motion of the system is given by

N

f i共 r 兲 ⫽

冋 册

T⫽⫺

共12兲

ប2 2␮

N⫺1

兺

j⫽1

2

ⵜ rជ ( j) ,

共17兲

␭

2

The rotating trial basis functions for diatomic molecules are given by f ␣ 共 x,y,z 兲 ⬅ f i 共 r 兲 Y lm 共 ␪ , ␾ 兲 ,

共13兲

where ⵜ rជ ( j) are Laplacians in three-dimensional 共3D兲 Carte␭

sian coordinates. At this point we propose a generalization of the procedure developed by Prudente and Acioli 关30兴. The initial set of basis functions is given by

where

˜f n , . . . ,n 1 N

r⫽ 共 x 2 ⫹y 2 ⫹z 2 兲 1/2,

␪ ⫽arccos

z 共 x 2 ⫹y 2 ⫹z 2 兲

, 1/2

共14兲

max

⫽exp

␯,␩

冊兿

␯ ⫽1

共 ⌬S ␯ 兲 n ␯ ,

共18兲

where N max ⫽N(N⫺1)/2, and ⌬S ␯ ⫽S ␯ ⫺S ␯0 with the local modes S ␯ defined by S ␯ ⫽ 储 rជ i ⫺rជ j 储

y ␾ ⫽arctan x are the spherical polar coordinates. Note that we needed these relations to calculate the overlap and Hamiltonian matrix elements.

冉兺

N max

⌬S ␯ A ␯ , ␩ ⌬S ␩

with 1⭐i⬍ j⭐N,

1⭐ ␯ ⭐N max .

共19兲

For three and four bodies we have 3N⫺6 independent internal coordinates and the same number of local modes; for larger systems the number of local modes is greater than the number of internal coordinates and we have an overcomplete

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PHYSICAL REVIEW A 61 012506

set of S ␯ 关23兴. Moreover, we can suppose that the potentialenergy surface has only one pronounced minimum and can define the values of the local modes there to be S 0␯ . The variational parameters 兵 A ␯ ␩ 其 are optimized in order to minimize the variational energy of the ground state 关in Eq. 共18兲 n ␯ ⫽0 ᭙ ␯ 兴 or its variance. Utilizing the VMC method, the new orthonormal vibrational basis set is f i 共 S 1 , . . . ,S N max 兲 ⫽

••• 兺 兺 n n 1

n •••n N max˜

N max

f n 1 •••n N 共 S 1 , . . . ,S N max 兲 ,

ci 1

共20兲

and, the rotating trial basis functions for the n-body system are given by f ␹ 共 r ␭( j)x ,r ␭( j)y ,r ␭( j)z 兲 ⬅ f i 共 S 1 , . . . ,S N max 兲 D JM N 共 ␣ , ␤ , ␥ 兲 , 共21兲 where D JM N is the Wigner function and ␣ , ␤ , and ␥ are the Euler angles. Note that there is no problem with the Coriolis terms because the Hamiltonian is written in 3(N⫺1)D Cartesian coordinates. Another important aspect is that the S ␯ local coordinates and the Euler angles ( ␣ , ␤ , ␥ ) have to be written as a function of the Cartesian components of the Jacobi vectors. The transformation of the local modes S ␯ to components of Jacobi coordinates is straightforward. However, this transformation for the Euler angles is not, and we need to perform some algebra and geometric manipulation; this is the object of next section.

To obtain the Euler angles in the Cartesian components of Jacobi vectors we start defining the 3⫻(N⫺1) Jacobi matrix

⫽

冉

uជ ␭(1)x

uជ ␭(2)x

...

uជ ␭(N⫺1)x

uជ ␭(1)y

uជ ␭(2)y

...

uជ ␭(N⫺1)y

uជ ␭(1)z

uជ ␭(2)z

...

uជ ␭(N⫺1)z

冊

,

共22兲

where uជ ␭( j)x , uជ ␭( j)y , and uជ ␭( j)z are the Cartesian components of rជ ␭( j) . We can take two of these vectors uជ ␭(k) and uជ ␭(l) , to determine one plane in R3 space, and choose a fixed refer-

U␭ ⫽R⫺1

冉

ence system O XY Z where the Cartesian components of these two vectors are

冉冊 冉 冊 r sin共 ␪ 兲

0

uជ ␭(k) ⫽

0

uជ ␭(l) ⫽

,

R

0

共23兲

,

r cos共 ␪ 兲

where R⫽ 储 uជ ␭(k) 储 , r⫽ 储 uជ ␭(l) 储 and cos(␪)⫽uជ ␭(k)•uជ ␭(l)/储uជ ␭(k)储 储uជ ␭(l)储. For the other N⫺3 vectors we have a general form in Cartesian components, such as uជ ␭( j) ⫽

V. EULER ANGLES IN CARTESIAN COMPONENTS OF JACOBI VECTORS

U␭ ⫽ 共 rជ ␭(1) . . . rជ ␭(N⫺1) 兲

FIG. 1. General scheme of the coordinate system for N particles. k and l are the labels of the two vectors used to define the Euler angles, and j⫽1, . . . ,N ( j⫽k, j⫽l) represents the remaining degrees of freedom.

冉

␳ j sin共 ␾ j 兲 ␳ j cos共 ␾ j 兲 sin共 ␦ j 兲 ␳ j cos共 ␾ j 兲 cos共 ␦ j 兲

冊

,

j⫽1, . . . ,N⫺1,

兵 j⫽k,l 其 , 共24兲

where ␳ j ⫽ 储 uជ ␭( j) 储 , ␾ j is the angle between the vector uជ ␭( j) and the Z axis of our system 共we note that this axis is defined by the vector uជ ␭(k) ) and ␦ j is the angle between the projection of the vector uជ ␭( j) in the XY plane and the X axis 共see Fig. 1兲. Now we can construct the system of equations that relates the Cartesian components of rជ ␭( j) in the fixed reference system given by the Jacobi matrix in Eq. 共22兲 with the general internal coordinates and the three external variables, the Euler angles, that describe the spatial rotations of the system. The system of equations has the following structure 共the second column is the k line and the third column is the l line兲:

␳ 1 sin共 ␾ 1 兲

...

0

...

r sin共 ␪ 兲

...

␳ N⫺1 sin共 ␾ N⫺1 兲

␳ 1 cos共 ␾ 1 兲 sin共 ␦ 1 兲

...

0

...

0

...

␳ N⫺1 cos共 ␾ N⫺1 兲 sin共 ␦ N⫺1 兲

␳ 1 cos共 ␾ 1 兲 cos共 ␦ 1 兲

...

R

...

r cos共 ␪ 兲

...

␳ 1 cos共 ␾ N⫺1 兲 cos共 ␦ N⫺1 兲

where R is the rotation matrix given by 012506-4

冊

, 共25兲

GENERALIZED ROTATING WAVE FUNCTION FOR . . .

R⫺1 ⫽

冉

PHYSICAL REVIEW A 61 012506

cos ␣ cos ␤ cos ␥ ⫺sin ␣ sin ␥

⫺cos ␣ cos ␤ sin ␥ ⫺sin ␣ cos ␥

cos ␣ sin ␤

sin ␣ cos ␤ cos ␥ ⫹cos ␣ sin ␥

⫺sin ␣ cos ␤ sin ␥ ⫹cos ␣ cos ␥

sin ␣ sin ␤

⫺sin ␤ cos ␥

sin ␤ sin ␥

cos ␤

冊

,

共26兲

and the system of equations is uជ ␭(1)x ⫽ ␳ 1 sin共 ␾ 1 兲 cos共 ␣ 兲 cos共 ␤ 兲 cos共 ␥ 兲 ⫺ ␳ 1 sin共 ␾ 1 兲 sin共 ␣ 兲 sin共 ␥ 兲 ⫺ ␳ 1 cos共 ␾ 1 兲 sin共 ␦ 1 兲 cos共 ␣ 兲 cos共 ␤ 兲 sin共 ␥ 兲 ⫺ ␳ 1 cos共 ␾ 1 兲 sin共 ␦ 1 兲 sin共 ␣ 兲 cos共 ␥ 兲 ⫹ ␳ 1 cos共 ␣ 兲 sin共 ␤ 兲 cos共 ␦ 1 兲 , uជ ␭(1)y ⫽ ␳ 1 sin共 ␾ 1 兲 sin共 ␣ 兲 cos共 ␤ 兲 cos共 ␥ 兲 ⫺ ␳ 1 sin共 ␾ 1 兲 cos共 ␣ 兲 sin共 ␥ 兲 ⫺ ␳ 1 cos共 ␾ 1 兲 sin共 ␦ 1 兲 sin共 ␣ 兲 cos共 ␤ 兲 sin共 ␥ 兲 ⫺ ␳ 1 cos共 ␾ 1 兲 sin共 ␦ 1 兲 cos共 ␣ 兲 cos共 ␥ 兲 ⫹ ␳ 1 sin共 ␣ 兲 sin共 ␤ 兲 cos共 ␦ 1 兲 , uជ ␭(1)z ⫽⫺ ␳ 1 sin共 ␾ 1 兲 sin共 ␤ 兲 cos共 ␥ 兲 ⫹ ␳ 1 cos共 ␾ 1 兲 sin共 ␤ 兲 sin共 ␥ 兲 sin共 ␦ 1 兲 ⫹ ␳ 1 cos共 ␾ 1 兲 cos共 ␤ 兲 cos共 ␦ 1 兲 , ⯗ uជ ␭(k)x ⫽R cos共 ␣ 兲 sin共 ␤ 兲 , uជ ␭(k)y ⫽R sin共 ␣ 兲 sin共 ␤ 兲 , uជ ␭(k)z ⫽R cos共 ␤ 兲 , ⯗ uជ ␭(l)x ⫽r sin共 ␪ 兲 cos共 ␣ 兲 cos共 ␤ 兲 cos共 ␥ 兲 ⫺r sin共 ␪ 兲 sin共 ␣ 兲 sin共 ␥ 兲 ⫹r cos共 ␣ 兲 sin共 ␤ 兲 cos共 ␪ 兲 , uជ ␭(l)y ⫽r sin共 ␪ 兲 sin共 ␣ 兲 cos共 ␤ 兲 cos共 ␥ 兲 ⫹r sin共 ␪ 兲 cos共 ␣ 兲 sin共 ␥ 兲 ⫹r sin共 ␣ 兲 sin共 ␤ 兲 cos共 ␪ 兲 , uជ ␭(l)z ⫽⫺r sin共 ␪ 兲 sin共 ␤ 兲 cos共 ␥ 兲 ⫺r cos共 ␪ 兲 cos共 ␤ 兲 , ⯗ uជ ␭(N)x ⫽ ␳ N sin共 ␾ N 兲 cos共 ␣ 兲 cos共 ␤ 兲 cos共 ␥ 兲 ⫺ ␳ N sin共 ␾ N 兲 sin共 ␣ 兲 sin共 ␥ 兲 ⫺ ␳ N cos共 ␾ N 兲 sin共 ␦ N 兲 cos共 ␣ 兲 cos共 ␤ 兲 sin共 ␥ 兲 ⫺ ␳ N cos共 ␾ N 兲 sin共 ␦ N 兲 sin共 ␣ 兲 cos共 ␥ 兲 ⫹ ␳ N cos共 ␣ 兲 sin共 ␤ 兲 cos共 ␦ N 兲 , uជ ␭(N)y ⫽ ␳ N sin共 ␾ N 兲 sin共 ␣ 兲 cos共 ␤ 兲 cos共 ␥ 兲 ⫺ ␳ N sin共 ␾ N 兲 cos共 ␣ 兲 sin共 ␥ 兲 ⫺ ␳ N cos共 ␾ N 兲 sin共 ␦ N 兲 sin共 ␣ 兲 cos共 ␤ 兲 sin共 ␥ 兲 ⫺ ␳ N cos共 ␾ N 兲 sin共 ␦ N 兲 cos共 ␣ 兲 cos共 ␥ 兲 ⫹ ␳ N sin共 ␣ 兲 sin共 ␤ 兲 cos共 ␦ N 兲 , uជ ␭(N)z ⫽⫺ ␳ N sin共 ␾ N 兲 sin共 ␤ 兲 cos共 ␥ 兲 ⫹ ␳ N cos共 ␾ N 兲 sin共 ␤ 兲 sin共 ␥ 兲 sin共 ␦ N 兲 ⫹ ␳ N cos共 ␾ N 兲 cos共 ␤ 兲 cos共 ␦ N 兲 . We observe that the equations for uជ ␭(k) and uជ ␭(l) form a subset of six equations for six variables, which are (R,r, ␪ , ␣ , ␤ , ␥ ). Solving this subset, we obtain the expressions for the three external variables 共Euler angles兲 as a function of the Cartesian components of a set of N Jacobi vectors that describe the n-body system. The solutions are given by

␣ ⫽arccos

␤ ⫽arccos

␥ ⫽arccos

冉

冉

冉

uជ ␭(k)x 关共 uជ ␭(k)x 兲 2 ⫹ 共 uជ ␭(k)y 兲 2 兴 1/2

冊

共27兲

,

uជ ␭(k)z 关共 uជ ␭(k)x 兲 2 ⫹ 共 uជ ␭(k)y 兲 2 ⫹ 共 uជ ␭(k)z 兲 2 兴 1/2

冊

uជ ␭(l)z 关共 uជ ␭(k)x 兲 2 ⫹ 共 uជ ␭(k)y 兲 2 兴 ⫺uជ ␭(k)z 共 uជ ␭(k)x uជ ␭(l)x ⫹uជ ␭(k)y uជ ␭(l)y 兲 ⌬ 1/2 012506-5

共28兲

,

冊

,

共29兲

COSTA, PRUDENTE, AND ACIOLI

PHYSICAL REVIEW A 61 012506

where ⌬⫽ 关共 uជ ␭(k)x 兲 2 ⫹ 共 uជ ␭(k)y 兲 2 兴关共 uជ ␭(k)x 兲 2 共 uជ ␭(l)y 兲 2 ⫹ 共 uជ ␭(l)z 兲 2 兴 ⫹ 共 uជ ␭(k)y 兲 2 关共 uជ ␭(l)x 兲 2 ⫹ 共 uជ ␭(l)z 兲 2 兴 ⫹ 共 uជ ␭(k)z 兲 2 关共 uជ ␭(l)x 兲 2 ⫹ 共 uជ ␭(l)y 兲 2 兴 ⫺2uជ ␭(k)x uជ ␭(l)x uជ ␭(k)y uជ ␭(l)y ⫺2uជ ␭(k)x uជ ␭(l)x uជ ␭(k)z uជ ␭(l)z ⫺2uជ ␭(k)y uជ ␭(l)y uជ ␭(k)z uជ ␭(l)z .

These expressions finalize the search for a generalized rotating wave function, because with Eq. 共21兲 we can construct this object for any number of particles. We observe that these expressions are not interesting from the point of view of the standard procedures to calculate the rovibrational spectra of n-body systems, where they search for an analytical solution of as many as possible degrees of freedom of the system. However, as stated in the Introduction, in four-body systems there is a tremendous difficulty in solving numerically this problem using conventional methods; for larger systems, it is impracticable, and we need to search for other methods. In particular we propose the utilization of the CFQMC method 共described in Sec. II兲 with the generalized rotating wave-function approach to implement the multidimensional calculations; for example, to study larger molecules such as H2 SiO 关34兴, clusters such as Hn ⫹ 关35,36兴 and Arn 关37兴, and van der Waals molecules such as ArHCN 关38兴.

This procedure utilizes rotating wave functions as a trial basis in the CFQMC method and its application to diatomic and triatomic molecules 关30,39兴 shown to be very efficient. We derived a generalization of n-body systems by defining the generalized rotating wave functions 关Eq. 共21兲兴. Moreover, one important relation for Euler angles as a function of Cartesian components of Jacobi vectors is obtained 关Eq. 共27兲兴. The main aspect that makes this approach possible is the fact that the quantum Monte Carlo method does not grow exponentially with the number of degrees of freedom and can be an alternative method of implementing numerical quantum calculations for systems with more than four atoms. In another work 关39兴 we applied the expressions developed in this work to calculated rovibrational energy levels of the water molecule for total angular momentum J⫽1,2 and obtained a good level of accuracy in the results.

VI. CONCLUSION

ACKNOWLEDGMENTS

In this work we have proposed one generalization for N atoms of the method proposed by Prudente and co-workers to compute the rovibrational levels of molecular systems.

The present work was supported by CAPES and CNPq through grants to the authors. The authors would like to thank Professor J. J. Soares Neto for helpful discussions.

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