A simple polynomial variational-perturbative approach to the evaluation of dynamic multipole polarizabilities for a ground-state hydrogen atom

5 October 2001

Chemical Physics Letters 346 (2001) 293±298

www.elsevier.com/locate/cplett

A simple polynomial variational-perturbative approach to the evaluation of dynamic multipole polarizabilities for a ground-state hydrogen atom Giuseppe Figari, Valerio Magnasco * Dipartimento di Chimica e Chimica Industriale dell'Universita', Via Dodecaneso 31, 16146 Genova, Italy Received 30 May 2001

Abstract Small well-shaped basis sets have been used to perform highly accurate variational-perturbative evaluations of dynamic (frequency-dependent) multipole polarizabilities for a hydrogen atom in its ground state. Their structure has been suggested by the ab initio polynomial power expansion we have recently developed to get correspondingly exact results within a ®nite range covering the region of low frequencies. The frequency-dependent coecients of the optimized linear combinations of the excited pseudostate orbitals were obtained by diagonalizing the representative matrix of the excitation operator from the ground state. High levels of accuracy have been observed over a set of frequencies now covering the complete real and imaginary ranges. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction We have recently suggested [1] a simple perturbative technique to get exact evaluations for dynamic (frequency-dependent) multipole polarizabilities of a hydrogen atom in its ground state. The ®rst-order perturbative correction to the unperturbed wave function (wf) was written by us as a product involving the eigenfunction of the square angular momentum operator corresponding to the given multipole, the exponential decay factor of the unperturbed wf and a radial term that has been expanded in powers of the frequency. Simple polynomial structures were found for the coecients of the expansion by applying an ele-

*

Corresponding author. Fax: +39-010-353-6199. E-mail address: [email protected] (V. Magnasco).

mentary approach that can be viewed as a straightforward extension of the algebraic procedure which gives the exact expression of the static (zero-frequency) multipole polarizabilities for the hydrogen atom in its ground state [2]. Although our polynomial technique is remarkably simpler than previous exact perturbative methods that have been developed to evaluate dynamic multipole polarizabilities of the hydrogen atom [3±7], its applicability is however restricted to a ®nite region of low frequencies, convergence for both real and imaginary frequencies being limited by the lowest excitation threshold for each required multipole. This is particularly inconvenient when the whole range of imaginary frequencies is required as in the evaluation of dispersion constants by the Casimir±Polder integration [8]. In this Letter, we want to test whether similar restrictions would a€ect the corresponding varia-

0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 8 1 6 - 8

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tional static±dynamic extension, keeping in the trial basis set the same simple polynomial radial dependence that has been obtained for the exact perturbed wf [1]. Thus the same basis sets were used to get variational values for frequency-dependent multipole polarizabilities. Optimized combinations of the excited orbitals included in each basis set were obtained by working, through simple matrix diagonalizations, on a straightforward dynamic extension of the static second-order perturbative functional. Relatively small bases have been found to be sucient to get highly accurate evaluations of frequency-dependent multipole polarizabilities of a hydrogen atom in its ground state, over the entire real and imaginary ranges. Our variational static±dynamic extension is therefore free from the restrictions that were observed for our corresponding exact expansion and we can propose it as a simple polynomial alternative to other variational techniques involving more elaborate basis sets and/or algebraic procedures [3,9±16]. 2. Approximate evaluation of the frequency-dependent polarizability by means of polynomial basis sets Atomic units (a.u.) are used throughout in this Letter. The frequency-dependent multipole polarizability a` …x† of a hydrogen atom in its ground state can be evaluated by using a suitable and well-known perturbative expression (see, for instance [17]). Its simple structure, where two matrix elements display a peculiar antithetic dependence on the given value of the frequency, can be derived through a straightforward application of the basic principles inherent to timedependent perturbative theory [18±20]: a` …x† ˆ hw0 jU` jw` …x†i ‡ hw0 jU` jw` … x†i …` ˆ 1; 2; 3; . . .†

…1†

where: ` is the angular quantum number corresponding to the given multipole; x is the given value p of the frequency; w0 ˆ exp… r†= p is the unperturbed 1s atomic orbital;

p U` ˆ 4p=…2` ‡ 1†r` Y`0 …#; u† is the multipole perturbative operator. Here r, #, u denote radial and angular co-ordinates belonging to a reference system centred on the nucleus of the hydrogen atom, and Y`0 …#; u† is the normalized eigenfunction with quantum numbers ` and 0 of the atomic square angular momentum operator L2 . A ®rst-order perturbative equation must be solved, exactly or at least in an approximate way, to get the explicit structure of the perturbed wf w` …x†: …H0

E0 ‡ x†w` …x† ˆ U` w0 :

…2†

The unperturbed atomic hamiltonian operator H0 is involved in the perturbative equation together with the unperturbed atomic energy E0 : H0 ˆ

1 o2 2 or2

1 o L2 ‡ 2 r or 2r

1 ; r

E0 ˆ

1 : 2

…3†

We assume that w` …x† can be faithfully represented by using an optimized linear combination of excited orbitals centred on the atomic nucleus. The orbitals /1 ; /2 ; . . . ; /N included in the excited multipole basis set and the corresponding frequency-dependent linear coecients C1 ; C2 ; . . . ; CN can be collected, respectively, into a row matrix / and column matrix C. w` …x† can thus be written as the matrix product: w` …x† ˆ /C:

…4†

To build a faithful variational counterpart to our previous exact perturbative procedure [1], we construct the radial polynomial part of the perturbed wf using radial powers …r` ; r`‡1 ; r`‡2 ; . . .† that are included together with the eigenfunction Y`0 …#; u† of the square angular momentum operator and the exponential decay factor of the unperturbed wf w0 : s 22`‡2j‡1 /j ˆ Y`0 …#; u†r` 1‡j exp… r† …2` ‡ 2j†! …5† …j ˆ 1; 2; . . . ; N †: The straightforward variational extension of our previous exact perturbative procedure is completed by replacing w` …x† with the corresponding row±column product /C within the di€erential

G. Figari, V. Magnasco / Chemical Physics Letters 346 (2001) 293±298

perturbative Eq. (2) and then by using the column /y to perform a premultiplication followed by integration over the entire space. Optimization of the linear coecients C yields the matrix equation: …M ‡ xS†C ˆ N:

…6†

While M, the matrix representative of the excitation energy operator H0 E0 , and S, the overlap matrix, are N  N square matrices, N is a N-term column matrix of transition multipole moments, their elements being: Mjk ˆ h/j jH0

E0 j/k i

`…` ‡ 1† ‡ …` 1 ‡ j†…` 1 ‡ k† ; ˆ 2Sjk …2` ‡ j ‡ k†…2` ‡ j ‡ k 1† …2` ‡ j ‡ k†! Sjk ˆ h/j j/k i ˆ p ; …2` ‡ 2j†!…2` ‡ 2k†! …2` ‡ j ‡ 1†! Nj ˆ h/j jU` jw0 i ˆ p ` 2 2…2` ‡ 1†…2` ‡ 2j†! … j ˆ 1; 2; . . . ; N ;

…7†

1

k ˆ 1; 2; . . . ; N †:

…8†

but a matrix inversion would also be required for any given value of the frequency x. We have therefore preferred to get an equivalent result through a single matrix diagonalization. After performing the diagonalization of the Hermitian square matrix M: Vy MV ˆ e …Vy SV ˆ 1†

with: e ˆ N  N diagonal square matrix of the eigenvalues (each corresponding to a peculiar excitation energy); V ˆ N  N square matrix of the eigenvectors; 1 ˆ N  N identity matrix; the square matrix of the eigenvectors can be used to transform the basis set / into the related orthonormal basis set /0 , usually known as pseudostate basis [21,22]: /0 ˆ /V:

…10†

The substitution of / with /0 allows us to get a completely equivalent approximate evaluation for the frequency-dependent multipole polarizability, provided by the diagonal form of matrix Eq. (6): …e ‡ x1†C0 ˆ l;

…11†

where l symbolizes the N-term column matrix collecting transition multipole moments, that has replaced N:

Although the static limit …x ! 0† of matrix Eq. (6) allows to get a lower bound for the corresponding static multipole polarizability, the same property does not hold (at least not generally) when the frequency is involved in the perturbative problem with any non-zero real or imaginary value. More sophisticated approaches [9,10,14] are required to provide lower and/or upper bounds for the dynamic multipole polarizabilities. Matrix Eq. (6) is therefore used [12,13,15,16] as a simple algebraic tool for yielding approximate solutions to the exact dynamic perturbative equation (2). It would undoubtedly be possible to evaluate the coecients of the linear set C by applying a simple matrix inversion: C ˆ …M ‡ xS† N;

295

…9†

l ˆ Vy N:

…12†

The general explicit and optimized formula for the linear coecients belonging to the set C0 will hence be given by: lj …j ˆ 1; 2; . . . ; N †: …13† Cj0 ˆ ej ‡ x Expression (1) for the frequency-dependent multipole polarizability a` …x† can accordingly be recast into the form: N h i X a` …x† ˆ hw0 jU` j/0j i Cj0 …x† ‡ Cj0 … x† jˆ1

ˆ

N X jˆ1

ˆ2

 lj

N X jˆ1

lj lj ‡ ej ‡ x ej x

ej l2j : e2j x2



…14†

The related expression for the multipole polarizability a` …iu† depending on imaginary frequencies can be readily obtained by replacing the real frequency x with the imaginary frequency iu (u standing for a real quantity): N X ej l2j a` …iu† ˆ 2 : …15† e 2 ‡ u2 jˆ1 j

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G. Figari, V. Magnasco / Chemical Physics Letters 346 (2001) 293±298

It is therefore possible to evaluate frequency-dependent polarizabilities of a ground-state hydrogen atom by performing sums-over-pseudostates where the contribution coming from the jth excited orbital /0j is speci®cally provided by its excitation energy ej (jth eigenvalue of the Hermitian square matrix M) and its transition multipole moment lj : ej ˆ

N X N X pˆ1 qˆ1

lj ˆ

N X

Vpj Mpq Vqj ˆ h/0j jH0

Vpj Np ˆ

pˆ1

E0 j/0j i; …16†

h/0j jU` jw0 i:

3. Numerical results and discussion To test the level of accuracy that can be achieved by applying our variational approach, we used progressively enlarged excited basis sets to perform perturbative evaluations of dynamic multipole polarizabilities for a hydrogen atom in its ground state, at given di€erent real and imaginary values of the frequency. Optimized estimates have been obtained, by means of expressions (14) and (15), for dipole …` ˆ 1†, quadrupole …` ˆ 2† and octopole …` ˆ 3† frequency-dependent polarizabilities.

Selected numerical results are reported in Table 1 (real frequencies) and Table 2 (imaginary frequencies), systematically underlining their unstable digits (i.e., the digits that could be modi®ed by enlarging the size of the excited basis sets involved in the calculation). The values of the real frequencies (Table 1) have been ®xed without exceeding their highest physically allowable value …x ˆ 0:5†. Since, on the contrary, no physical upper limit holds for the imaginary frequencies, a sensibly larger range of values can be involved in these calculations whose results are collected in Table 2. In both tables, a simple inspection of the decreasing trend displayed everywhere by the underlined unstable digits allows to appreciate the reliability of our variational procedure: pseudostate basis sets of small or moderate size are always sucient to yield a satisfactorily large number enough of stable exact digits (stable digits of our present results are also exact digits resulting from exact power expansions [1,4] of the corresponding dynamic polarizabilities). We notice that smaller and smaller basis sets are needed to get accurate evaluations for dynamic multipole polarizabilities corresponding to progressively smaller values of the real or imaginary frequencies involved in the calculation. This pecu-

Table 1 Dynamic multipole polarizabilities a` …x† of a ground-state hydrogen atom, from variational calculations performed at given real frequencies x by using N-term pseudostate basis sets (unstable digits are underlined) N

a1 …x†=a30

a2 …x†=a50

a3 …x†=a70

x=Eh ˆ 0:10 5 10

4.784 300 343 4.784 300 343

15.519 764 399 15.519 764 399

134.679 646 528 134.679 646 529

x=Eh ˆ 0:20 5 10

5.941 674 179 5.941 674 861

17.367 109 547 17.367 110 026

146.370 653 675 146.370 655 096

x=Eh ˆ 0:30 5 10 15

10.563 260 756 10.563 888 866 10.563 888 867

22.014 486 644 22.014 652 591 22.014 652 591

172.568 895 340 172.569 259 070 172.569 259 070

x=Eh ˆ 0:40 5 10 15 20 25

)18.249 )16.824 )16.822 )16.822 )16.822

40.307 40.462 40.462 40.462 40.462

241.173 241.298 241.298 241.298 241.298

583 209 645 645 645

751 094 917 355 354

856 601 689 689 689

832 200 588 613 613

633 600 639 639 639

077 762 659 667 667

G. Figari, V. Magnasco / Chemical Physics Letters 346 (2001) 293±298

liar trend is not unexpected, since a linear combination of just the ®rst two excited orbitals of our basis sets is necessary to provide the exact perturbed wf w` related to the static multipole polarizability [2] (i.e., to the zero-frequency common limit of the dynamic multipole polarizabilities depending on both real and imaginary frequencies):  `  2 r r`‡1 w` ˆ p Y`0 …#; u† ‡ exp… r†: ` `‡1 2` ‡ 1 …17† This also explains why an increasingly ®ner partitioning of static polarizabilities is needed for increasing the accuracy in the evaluation of dispersion constants for the interaction of two or

297

three ground-state H-atoms [23,24]: a larger number of static pseudostate contributions in the generalized London formula [23±25] is the equivalent of a larger number of frequency contributions in the Casimir±Polder integral [8]. More surprisingly, basis sets of remarkably small size turn out to be appropriate when dealing with extremely high values of the frequency. This unexpected behaviour can be ascribed to the peculiar character of the contribution coming from the ®rst orbital included in our excited basis sets. The estimate provided for the dynamic multipole polarizability at imaginary frequency a` …iu† by a 1term basis set containing only our ®rst excited orbital, has the simple explicit expression:

Table 2 Dynamic multipole polarizabilities a` …iu† of a ground-state hydrogen atom, from variational calculations performed at given imaginary frequencies iu by using N-term pseudostate basis sets (unstable digits are underlined) N

a1 …iu†=a30

a2 …iu†=a50

a3 …iu†=a70

u=Eh ˆ 0:10 5

4.250 298 265

14.518 589 616

128.016 906 492

u=Eh ˆ 0:20 5 10

3.653 832 603 3.653 832 590

13.263 803 160 13.263 803 142

119.328 603 277 119.328 603 212

u=Eh ˆ 0:40 5 10

2.381 754 068 2.381 754 587

9.975 786 218 9.975 786 553

94.675 887 814 94.675 887 963

u=Eh ˆ 5:00 5 10 15 20 25

0.039 0.039 0.039 0.039 0.039

016 017 017 017 017

993 359 163 166 166

0.232 0.232 0.232 0.232 0.232

677 671 671 671 671

608 289 020 025 025

2.611 2.611 2.611 2.611 2.611

330 267 267 267 267

086 962 431 446 445

u=Eh ˆ 20:00 5 10 15 20 25 30

0.002 0.002 0.002 0.002 0.002 0.002

495 494 494 494 494 494

356 482 452 461 461 461

0.014 0.014 0.014 0.014 0.014 0.014

968 966 966 966 966 966

053 493 426 434 434 433

0.168 0.168 0.168 0.168 0.168 0.168

374 368 368 368 368 368

986 961 760 770 770 770

u=Eh ˆ 100:00 5 10 15 20

0.000 0.000 0.000 0.000

099 099 099 099

992 990 989 989

0.000 0.000 0.000 0.000

599 599 599 599

948 945 945 944

0.006 0.006 0.006 0.006

749 749 749 749

396 385 384 383

u=Eh ˆ 200:00 5 10 15

0.000 025 000 0.000 024 999 0.000 024 999

0.000 149 997 0.000 149 997 0.000 149 997

0.001 687 462 0.001 687 462 0.001 687 461

298

G. Figari, V. Magnasco / Chemical Physics Letters 346 (2001) 293±298

a` …iu† ˆ 2 ˆ

References

e1 l21 M11 N12 ˆ 2 2 ‡ u2 M11 ‡ u2

e21

`…2`†!

22`

1

h i; `2 =…` ‡ 1†2 ‡ u2

…18†

whose asymptotic form in the limit of high frequencies: ! `…2`†! `2 2 u  a` …iu† ˆ 2` 1 2 …19† 2 2 u …` ‡ 1† is just the leading term arising from the asymptotic exact expansion of a` …iu† in reciprocal powers of the given imaginary frequency iu [4,26]. Comparison with Tables 1 and 2 of our previous Letter [1] shows that the number of variational terms in the radial powers required to get a 9decimal digit accuracy at x=u ˆ 0:4 is exceedingly smaller than the number of terms in the frequency powers required in the exact calculation (25 for x, 10 for u compared to 120 terms for both frequencies, and excluding the dipole term). We can therefore conclude that our simple polynomial variational approach can be pro®tably used to yield highly accurate evaluations of frequency-dependent multipole polarizabilities of a ground-state hydrogen atom over the entire real and imaginary ranges. Acknowledgements Financial support by the Italian Ministry for University and Scienti®c and Technological Research (MURST) under Grant No. 9803246003 and by the Italian National Research Council (CNR) is gratefully acknowledged.

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