Complex resonance eigenvalues by the lanczos recursion method

Volume 130, number 1,2

26 September 1986

CHEMICAL PHYSKS LETTERS

COMPLEX RESONANCE EI~ENVALUES

BY THE LANCZOS RECURSION METHOD *

Kent F. MILFELD and Nimrod MOISEYEV ’ Department of Chemtstry and Institute for Theoretical

Chemutry.

The Umverslty of Texas, Austm, TX 78712, USA

Received 14 May 1986; in final form 1I July 1986

A complex (non-standard) Lanczos recursion method (LRM) is used to determine resonance positions and widths from a Hamiltonian by the complex coordinate method (CCM). When the initial Lanczos vector is an approximate resonance eigenvector (e.g. obtained by the Taylor-Hazi stabilization method) the resonance positions and widths of a NxN Hamiltonian matrix are obtained by the diagonallzation of a much smaller m x m tridiagonal matrix ( m Q N). The combination of the complex LRM and CCM to determine complex resonance enetgtes is well suited for large Hamiltonian matrices:: providing faster execution and requiring less memory than standard methods. A numerical example is presented.

1. Introduction The complex coordinate method [ 1,2 ] (CCM) has been used to find accurate resonances in atomic [ 3 ] and molecular autoionization [ 4,5 1, vibrational predissociation [ 6-81, van der Waals complexes [ 9- 111, and recently, gas-surface scattering [ 121. Autoionization and predissociation resonances are metastable states embedded in a continuum. The energies (E) and widths (T ) of resonance states are derived from the complex eigenvalues of the time-independent Schrijdinger equation. By analytically continuing the Hamiltonian into the complex plane, the energy position and lifetime of the resonance states are realized as the real part and two times the reciprocal of the imaginary part of the complex energy (t=Eif/2). In the complex coordinate method both the bound and resonance states are represented by square integrable functions, f Y >. The internal coordinate, r, of the Hamiltonian operator is continued into the complex plane (also referred to as a rotation by the angle 0): that is, H(r)+H(re’“)=H(B). If 8 is large

enough, then the wavefunction of a resonance state vanishes as 1t-1+ co and Y resis given by, y,,,= CC&(r)

,

(1)

where the set of basis states {@,eL2} is complete. In numerical calculations a resonance is found by looking at the complex spectra of the Hamiitonian matrices H ( 0) at various angles. Resonance energies remain almost constant over a wide range of B [ 11,13,14 ] (provided a large enough basis is used); whereas the eigenvalues associated with the pure continuum state change markedly over this range. The complex analog of the variational p~nciple [ 61 gives the criteria for determining exact resonance eigenvalues:

at/de=0and adac,=o,

(2)

where E is a complex eigenvalue of the rotated Hamiltonian and the Cs are expansion coefficients of the wavefunction (see eq. (1) ). The second differential in eq. (2) leads to the matrix eigenvalue problem, HC= CC,

(3)

* Supported in part by ETA Systems, Inc., 1450 Energy Park, St. Paul, MN 55 108, USA and the National Science Foundation.

where

’ Permanent address: Department of Chemistry, Technion -Israel Institute of Technology, Haifa 32000, Israel.

0 009-2614186 $03.50 OElsevler Science Publishers B.V. ~o~-Ho~~d Physics Pub~s~ng Division)

&=<@,IH(QI@).

(4) 145

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Note that H,,=H,, and, therefore, the Hamiltonian matrix is symmetric. Using the second criterion, de/a6 =O, the stationary solutions are realized as cusps in &trajectories [ 151 (in plots of E versus 6). With a small basis a stationary point may be a narrow cusp; however, as the basis set is increased its complex resonance eigenvalue becomes less affected by 8, as expected on the basis of the Balslev-Combes theorem [ 161. (See refs. [ 1 l- 131 for numerical illustrations of this phenomenon. ) The ability to determine resonances in excited multimode molecules, or scattering events with many channels and target states is not governed solely by the computational resources of the investigator, but also by the algorithm which determines the resonances. Standard diagonalization methods [ 171, which accurately determine the entire spectrum of the Hamiltonian are approximately N3 type processes, i.e. if N is the order of the matrix then over N3 arithmetic operations are required for the solution. These methods also require the Hamiltonian matrix to be in core. The Lanczos algorithm can determine accurate resonance energies in = N2 x m ( m = number of recursions) operations [ 18,191. More importantly, it is not necessary to retain the whole matrix in core. Segmentation of the algorithm for parallel processing, or serial processing on a single processor is easily implemented. Each segment only operates on a section of the Hamiltonian, allowing the Hamiltonian to be much larger than the size of the central memory of the machine. The purpose of this paper is to show that a complex extension of the Lanczos algorithm is straightforward, and its application to complex coordinate Hamiltonians is well suited for determining the complex energies of resonances. We present these tindings in the following sections. In section 2 we describe the application of the Lanczos method to the complex (rotated) Hamiltonian. In section 3 the method is tested using a modified recursive Lanczos procedure which was initially developed for real symmetric matrices. Resonances of a simple onedimensional model Hamiltonian are determined and benchmarked. Comparisons with exact results and convergence tests illustrate the merits of this new procedure. 146

LETTERS

26 September

1986

2. Application of the Lanczos method to complex rotated Hamiltonians A detailed development of the Lanczos recursion method as well as a discussion in more mathematical terms can be found elsewhere [20]. The Lanczos method has been implemented by Moro and Freed [ 2 11, and by Haller, Koppel and Cederbaum [ 221 to determine eigenvalues of diagonally-complex symmetric matrices with the aid of a complex-valued pseudo-scalar product; and a more mathematical discussion of the complex-symmetric matrix problem is given by Ruffinatti [ 231, and by Cullum and Willoughby [ 241. We also make use of the pseudoscalar product method in the Lanczos recursion method to determine resonance energies and widths which are associated with complex eigenvalues of a complex rotated Hamiltonian. We present a brief development for the purpose of clarity and understanding. The Lanczos recursion method generates a m x m tridiagonal matrix, T, by constructing an orthogonal matrix, U, which forms T by the similarity transform U- ’ HU. The eigenvalues of the matrix T are also eigenvalues of the NxN square general matrix H. Since the complex rotated Hamiltonian is represented by a complex symmetric matrix, its tridiagonal matrix is also symmetric as well. From the similarity transformation one can see that H(@[u,,uz,

. ..u.l=[u,,u2,

. ..u.lT,

(5)

where {u,, i= 1, . ... m} are orthonormal column vectors of U, N is the order of H (6) and the only nonzero matrix elements in T are T,,, + , = T,, +,,n =/I,,, and T,,, = a,.By considering the sequence of vector multiplications, u,, u2, ... ontoH(e) andTineq. (5) itis easy to see that H(Q%=B,-l&-i

.

+%l~,+P,u,+,

(6)

Apart from an indicial equation which must be supplied with an initial vector, the recursion involves only two adjacent II vectors at a time. Rearrangement of eq. (6) gives the residual vector P,, and the operational recursion formula, P n+I=[H(e)-a,l~,-Bn-rUn-*

9

from which each successive off-diagonal

(7) matrix ele-

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CHEMICAL PHYSICS LETTERS

ment /I,, = T,, L,nand vector u,&+1are constructed:

u n+ I =Pn+ 1% .

(91

The diagonal matrix elements, on= T,,.,, are obtained from the projection on=(g;,

H(@)u,) .

(10)

(Note that in principle it is possible that (uz, u, ) = 0 and sn #O for a bio~hogonal vector. However, rounding errors in the computations will always allow a no~alization such that (MT,u, ) = 1.) If the algorithm is initialized with vector aI and N- 1 (N is the order of H) recursions are performed in exact arithmetic, then all of the eigenvalues of 7 are identical to those of H as obtained by standard eigenvector-eigenvalue methods. By using an alternate initial vector, and recurring for less than N- 1 times, particular eigenvalues will still converge to their exact values. The initial vector plays a key role in what the spectrum of T represents [ 25 1. It has been shown that the Lanczos recursion method is a special case of a broader theory of moments [26]. It is easy to conclude that, just as in moment methods, the Lanczos recursion method generates an ~~imensional subspace of H which is directly dependent on the initial vector. That is, a subspace is generated in which the initial vector resides. It has been shown by Wyatt and co-workers f 19,25,27,28] that the residues generated from the spectrum of T yield all information for the dynamics of the initial vector (all the non-zero components of the eigenvector projections onto the initial vector are obtained frequently after only m < N recu~ions). How quickly an eigenvalue in the spectrum of T converges as m is increased not only depends on the particular Hamiltonian, but also on the character of the initial vector. As an extreme case, if the initial vector is an eigenvector of H, then the first recursion solution yields an exact eigenvalue of H. Therefore, in order to minimize the number of the recursions in the Lanczos method and thereby reduce the length of the computation, the initial vector should be an estimate of the resonance state. Since resonance positions can be estimated in most cases by diagonalizing a relatively small matrix (see, for

26 September 1986

example, the stab~ization method by Taylor and Hazi [ 291, and Holoein and Midtall 30]), not much effort is required to obtain an initial vector with the characteristics of a resonance state. As an example: Let H (@) be real by making 8 = 0, and use a small basis set to construct the matrix of this Hamiltonian. Determine the approximate position of a resonance by observing the stability of its eigenvalues as the basis size is increased. Then use the real eigenvector components of the approximate-resonance eigenvalue as the non-zero components of an initial vector for the Lanczos recursion. Adaptation of the Lanczos recursion method to obtain complex resonance eigenvalues of a rotated Hamiltonian is st~i~tfo~ard. The procedures are the same except for the following: (1) Rather than using the Lanczos procedure for non-~e~itian matrices [ 201, use the recursive procedures which bring a real symmetric matrix to tridiagonal form. As pointed out previously, allow the recursive vectors u, and the parameters (Y,and j3nto be complex. Also, use the pseudo-scalar dot product, u;f*l-l,, throu~out, i.e. do not conjugate the complex components of u, in the product. ( 2) From a small, complex rotated (or real) matrix find an approximate resonance eigenvector using standard diagonalization techniques. Use the components of this vector as the non-zero components of the initial vector for the Lanczos algorithm. The resonance character of this approximate resonance state can be evaluated by the Taylor-Hazi stabilization method [ 291. (3) Repeat the Lanczos algorithm several times with the same initial vector, increasing the number of basis functions with each repetition. An exact resonance position, E, and width, r, are obtained from the converged complex Lanczos eigenvalue, c=E-irf2.

3. Numerical example In this section the complex Lanczos algorithm for calculating resonance eigenvalues is demonst~ted. The one-dimensional model H=-~d2/dx2+(0.5~*-0.8)

(11)

xexp( -0.1x*)+0.8 147

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CHEMICAL PHYSICSLETTERS

was choosen as the Hamiltonian because of its use as a testbed for CCM theories. Previous CCM studies have demonstrated variational methods [ 6 ] , as well as the existence of upper and lower bounds for resonance positions and widths [ 31-331 with this Hamiltonian. Harmonic oscillator functions were used to construct the Hamiltonian matrix according to the method proposed by Harris and co-workers 1341, which is equivalent to the Gaussian quadrature procedure [ 351. The coordinate x in eq. ( 11) was scaled by the factor r,r=exp ($3). Energy units are in atomic units (h was scaled to unity). Since stabilization techniques [ 61 have already characterized the resonance at position E FS2.12, this resonance was choosen as the first state to be tested. By using a basis of only ten states, setting 0 to zero [ H( 13~0.0; IV= lo)] and diagonalizing H by standard methods a reasonable estimate for the resonance was found among the eigenvectors. The Hamiltonian matrix for the Lanczos algorithm was constructed with 60 harmonic oscillator functions as described before, and the angle 8 set to 0.4 [ H (8 = 0.4; N= 60)]. After m recursions the eigenvalues of the tridiagonal T were determined and scanned for values close to E = 2.12. For several values of m the eigenvalues of the resonance state are listed in table 1. After 52 recursion steps the Lanczos procedure converged to eleven significant digits. Convergence to six digits of precision was obtained after only 32 recursions. Only a slightly quicker convergence was obtained by using an initial vector from the Hamiltonian H(B=0.4; N= 10). The advantage of taking an approximate-resonance eigenvector (even with 8 =O) for the initial recursion vector, as opposed to an arbitrary vector with unifo~ com~nents, can be seen by comparing the results in tables 1 and 2. The conditions for the calculations in table 2 are identical to those of table 1, except that the initial uniform vector, U, = (l,l, ...)1)/If”*, was used to obtain the convergence sequence in table 2. Many more recursions (more than 55 ) were required to achieve a converged resonance eigenvalue. As seen from tables 1 and 2 it is necessary to increase the number of recursions until the eigenvalues of H (B; N) converge to acceptable precision. Fig. 1 shows the dependence of m on N for two different 148

26 September1986

Table 1 Convergence of Lanczos energy e for H (6 = 0.4; N= 60). IB, is the “resonance” fromH(B=0.0;N=lO)

HamiltonIan matrix eigenvector obtained

Number of recursions m

Ret

Imt

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

2.141027 2.128113 2.128057 2.127144 2.127137 2.127154 2.127184 2.127185 2.127186 2.127191 2.127196 2.127196 2.127197 2.127197 2.127197

-0.016407 -0.ot4019 -0.014012 -0.013966 -0.015390 - 0.015408 -0.015454 -0.015446 -0.015442 -0.015443 -0.015447 -0.015447 -0.015447 -0.015447 -0.015447

Complex

resonance

eigenvalue

Table 2 Convergence of Lanczos eigenvalue, t, as a function steps, m. H(0=0.4, N=60). The components (~,),=(1/10)“*fori=1-10and(u,),=0.0forr=11-60

of recursion of II, are:

Number of recursions m

Complex Ret

Ime

26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

2.108289 2.121769 2.120829 2.120604 2.120794 2.121235 2.122356 2.126564 2.125654 2.125420 2.126333 2.125933 2.127117 2.127209 2.127197 2.127197 2.127197 2.127197

0.029227 0.014550 -0.000779 -0.003336 -0.004637 -0.007447 -0.016567 -0.016086 -0.014665 -0.014459 -0.013635 -0.018951 -0.015550 -0.015459 -0.015448 -0.015447 -0.015447 -0.015441

resonance

eigenvalue

CHEMICAL PHYSICS LETTERS

Volume 130, number I,2

40

10

Re3 ”

Y

10

I

I

40

50

8

I

0

30

20

,

60

MATRIX ORDER (N)

Fig. 1. The number of recurstons, m, required to achteve convergence to three and eleven srgniticant figures as a function of N, the number of basis functions.

accuracy criteria. The important feature is that even for larger matrix orders the number of recursions required to achieve convergence is smaller than N. We have also found this dependency to be typical of several other resonances. The 2.12 au resonance is neither a narrow nor broad resonance compared to the others found in the spectrum. Very broad resonances are diffiuse and

26 September 1986

therefore require a large basis set to describe their asymptotic behavior. Very narrow resonances have very small widths, r, orders of magnitude smaller than the energy position, E, high-energy basis functions are needed to accurately describe their “dampened” wave form. Therefore, both narrow and broad resonances require larger Hamiltonian matrices. Table 3 lists the complex Lanczos eigenvalues of both a narrow and a broad resonance as a function of N (the broad resonance is aboue the potential barrier). At a matrix order of 300 the narrow resonance eigenvalue has converged to nearly eleven significant figures and the broad resonance value to eight significant figures. Note that the narrow resonance converges only to a precision of the least significant figure of the real part of the energy; this has also been observed previously by Christoffel and Bowman [361. The above results have characterized the convergence of the complex eigenvalues for various matrix orders and recursions, N and m. The value of B in H (8; N) may influence the speed of the convergence to the true resonance energy and width. N-trajecto~ plots are used to show the spiraling convergence of the complex eigenvalues as N, the number of the basis functions, is increased. Fig. 2b shows e-trajectory plots for three different rotation angles: 8=0.2, 0.3 and 0.4. For any choice of 8 the es converge to the same value, but at different rates. The 8 = 0.4 trajeo tory begins (N=20) with a much better estimate of the converged value of e, and also converges with fewer basis functions. Fig. 2b shows a blowup of the 8=0.4 trajectory of fig. 2a. Here, too, the Lanczos method shows a smooth path to convergence.

Table 3 Convergence of a narrow and a broad resonance energy by the complex Lanczos method. The initial vectors were derived from N= 100) for each matrix, H(0~0.4; N); NX 3/4 recursions were taken Matnx order N

Narrow resonance Ret

Imrx104

Re e

Imt

100 125 150 175 200 250

1.420970950 1.42097095 1 1.42097095 1 1.42097095 i 1.42097095 1 1.42097095t

-0.5826651 -0.5826681 -0.5826693 -0.5826699 -0.5826671 -0.5826670

3.25548421 3.25548680 3.25548657 3.25548635 3.25548629 3.25548627

-1.11115454 -1.11153251 -1.11153158 -1.11153162 -1.11153161 -1.11153159

H( 0 =

0.4;

Broad resonance

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4. Summary

-00152-

-0

PHYSICS

d

o15d

-00154

[0=031

-00154

[8=041 2.1270

..r-.(N=30-100) se . I 21272

I 21271 REAL

(N=30-100) 212;3

t‘

We have shown that the Lanczos recursion method (LRM) can be used to efficiently determine complex resonance positions and widths for Hamiltonians which support metastable states. After analytically continuing the Hamiltonian into the complex plane according to the complex coordinate method (CCM), resonances are realized as the stationary energies of complex Hamiltonian matrices. A complex Lanczos algorithm is used to find specific resonances of interest. The algorithm follows the same procedures used for Hermitian matrices; however it is extended to the complex plane, and the complex pseudo-scalar product U;U, (rather than the real product ~,**a,) is used to evaluate the bracket ( u,,~u,) as other investigators have done [21-241. The application of the Lanczos recursion method to obtain resonance positions and widths of complex-rotated Hamiltonians is new. Very accurate positions and widths for both narrow and sharp resonances were obtained for a well known one-dimensional model which has been used in previous investigations as a testbed for new CCM theories and computational procedures. The method also has the advantage of providing the resonance energies through the diagonalization of a tridiagonal matrix which is much smaller than the original Hamiltonian matrix (e.g. see fig. 1). This feature, as well as the other characteristics of the Lanczos procedure, makes this method suitable for investigating resonances in large Hamiltonian matrices which cannot be diagonalized by conventional procedures.

Acknowledgement

2.1271970

21271971 REAL

2.1271972

R.E Wyatt

for his

r

Fig. 2. Spiraling convergence of the resonance eigenvalue with increasing number of basis functions, N. (a) Three N-trajectories; 0=0.2, 0.3 and 0.4 from top to bottom, respectively. (b) Blowup of the 0 = 0.4 N-trajectory.

150

We are grateful to Professor helpful discussions.

References [ 11 W.P. Reinhardt,

Ann. Rev. Phys. Chem. 33 (1982) 223. [2] B.R. Junker, Advan. At. Mol. Phys. 18 (1982) 207. [ 31 Intern. J. Quantum Chem. 14 (1978)4. [ 41 N. Moiseyev and C.T. Coraran, Phys. Rev. A20 (I 979) 8 14.

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[S] S. Yabushita and C.W. McCurdy, J. Chem. Phys. 83 (1985) 3547, and references therein. [6] N. Moiseyev, P.R. Certain and F. Weinhold, Mol. Phys. 36 (1978) 1613. [ 7) R. Lefebvre, Chem. Phys. Letters 70 (1980) 430. [S] N. Moiseyev, Intern. J. Quantum Chem. 20 (1981) 835. [ 91 Z. Basic and J. Simons, Intern. J. Quantum Chem. 14 (1980) 467. [lo] S.I. Chu, J. Chem. Phys. 72 (1980) 4772. [ 111 N. Moiseyev and P.R. Certain, J. Phys. Chem. 89 (1985) 3853. [ 121 N. Moiseyev, T. Maniv, R. Elber and R.B. Gerber, Mol. Phys. 55 (1985) 1369. [ 131 B.R. Junker, Intern. J. Quantum Chem. 14 (1978) 371. [ 141 P. Froelich, 0. Goscinski and N. Moiseyev, J. Chem. Phys., to be published. [ 151 N. Moiseyev, S. Friedland and P.R. Certain, J. Chem. Phys. 74 (1981) 4739. [ 161 E. Baisiev and J.M. Combes, Commun. Math. Phys. 22 (1971) 180. [ 171B.N. Partlett, The symmetric eigenvaiue problem (Prentice-Hall, En&wood Cliffs, 1980) ch. 8. [ IS] C. Lanczos, J. Res. Natl. Bur. Std. 45 (1950) 255. [ 191 R.E. Wyatt and D.S. Scott, in: Large eigenvalue problems, eds. J. Cullum and R. Willoughby (North-Holland, Amsterdam, 1986). 1201 J.H. Wilkinson, The algebraic eigenvahte problem (Oxford Univ. Press, Oxford, 1968) p. 388. [ 2 I ] K.F. Milfeld, J. Castillo and R.E. Wyatt, J. Chem. Phys. 83 (1985) 1617.

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[22] Yu.V. Vorobyen, Method of moments in applied mathematics (Gordon and Breach, New York, 1965). [23] G. Moro and J.H. Freed, J. Chem. Phys. 74 (1981) 3757. [24] E. Haller, H. Koppel and L.S. Cederbaum, J. Mol. Spectry. 111 (1985)377. (25 ] J.G. Ru~natti, Computing 15 (1975) 275. [26] J.K. Cullurn and R.A. WiIIou~by, Lanczos algorithms for large symmetric etgenvalue computations, Vol. I (Theory) (Birkhluser, Basel, 1985). [ 271 A. Nauts and R.E. Wyatt, Phys. Rev. Letters 5 I (1983) 2238: Phys. Rev_ A30 (1985) 872; J.E. Castillo and R.E. Wyatt, J. Comput. Phys. 59 (1985) 120. [28] 1. Schek and R.E. Wyatt, J. Chem. Phys. 83 (1985) 3028, 4650. [29] H.S. Taylor and A.U. Hazi, Phys. Rev. A14 (1976) 2071. [ 301 E. Ho&n and J. Midtal, J. Chem. Phys. 45 (1966) 2209. [ 3 I J N. Moiseyev, in: Lecture notes in physics, Vol. 2 I 1, eds. S. Albereio, L.S. Ferreira and L. Streit (Springer, Berlin, 1984) p. 285. [ 321 N. Moiseyev, P. Froelich and E. Watkins, J. Chem. Phys. 80 (1984) 3623. [ 331 E.R. Davtdson, E. Engdahl and N. Moiseyev. Phys Rev, A, to be published. [ 341 D.O. Harris, H.W. Harrington, A.L. Luntz and W.P. Gwinn, J. Chem. Phys. 44 (1966) 3467. [35] A.S. Dickinson and P.R. Certain, J. Chem. Phys. 49 (1968) 4209. [ 361 K.M. Cnstoffel and J.M. Bowman, J. Chem. Phys. 78 (1983) 3952.

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