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Polynomial expansions for spectral densities
POL?JNOMLU
1 August 1976
CHEMICAL PHYSICS LETTERS
Volume 41, number 3
EXPANSIONS
FOR SPECTRAL
DENSlTLES
C.T. CORCORAN* Department of Ci’zemistry. Indiana University, Bioomìngron,
Indiana 47401, USA
and
P.W. LAXGHOFF** Department o_fChemistry, Indiana University, Bkwmington, Indìana 47401, USA and Department of Chemìstry, Stanford Universìty, Stanford, Gzlifornia 94305. USA and Joint Institute for Laboratory Astrophysìcs, Natìonal Bureau of Standardsand Universìty of Glorado. Boulder, Colorado 80309, USA Received 12 April 1976
Modified moments and polynomial expansions are employed in imaging scattering and photoionïzation
the spectra1 densities that arise in electron-
calculations.
1. lntroduction
rotation techniques in the latter connection. me theory of moments and Stieltjesintegration have al-
electron-scattering cross sections [lV] . Altllough the various approaches reported are satísfactory*, generalïzations and alternative perspectíves are always useful. In the present letter, modifïed moments and polynomíal expansions [20,21] are employed in approximating the spectral densítíes that aríse ín electronscattering and photoionization calculations. A general technique ís thereby provided for performing scattering and photoionization calculations in Hilbert space for an arbítrary many-electron system, employíng arbitrary basis sets of L2 functions. In the case of appropriate oneelectron hamiltonians and Laguerre or oscillator basis functions, the technique gives a generalization of the previously reported equivalentquaclrature approach [5-71, providing an approximate, convergent spectra1 densíty over a continuous range of energies. Moreover, when the density is eval-
so been employed in L2 calculations of photoabsorption and díspersion profdes [ 13-1 S] and elastic
* A consïderable number of publications dealing wïth the use
Considerable attention has focused recently on calculations of electron-scattering and photoionization cross sections entirely in Hilbert space. Reinhardt and co-workers have reported Lz calcuìatíons of Fredholm determinants El-31 and dípole polarizabilitíes [4] at complex energies and their analytic continuation to the real axis, and more recently have described an L2 equivalent-quadrature technique from which phase shifts and photoionization cross sections are obtained directly [S-71. In a separate but relatec! development, Heller and Yamani [8-101 have described a J-matrix approach to scattering and photoionization calculatíons, and McKoy and cowerkers [ 11,123 have reported the use of coordinate-
* Proctor and Gamble Fellow. ** On leave of absente from Indiana University, formerly visiting Stanford Unïversity, and presently JILA Visiting Fellow.
of square-integsable basis functions in scattering and other continuum calculations has appeared recently. The develop ment reported in the present letter is perhaps most closely related to the pubkations clterl in refs. [l-191, and no attempt is mzde hese at a complete citation of the recent literature in this area.
609
Volume 41, number 3
uated at the appropriate quadrature abscissa, results identical to the equivaIentquadrature technique are obtained, indicating that convergente of the latter approach can be estabhshed employing the rheory of moments. Polynomial expansions for the spectral densities that arise in scattering and photoionization calculations are descrïbed in sections 2 and 3, and connections with the equivalentquadrature technique are ìndicated. fllustrative apphcaiions Li the cases of Tlectron-hydrogen elastic scatterin& and photdonization in the negative hydrogep ion are gïven in section 4. The rapid convergente obtained in these exarrp!es suggests that the polynomial-expansion approach for imaging spectra1 densities should provide a useful addition tc the growing list of techniques for performing scattering and photoionization calculations entirely in Hilbert space. Some concluding tema& are gíven in sectiori 5.
2. Spectra1 densities The mean vaIue of a resolvent test function a, I$(z)
(W-
e)$c =o,
(1) Hhas
a continuous
O
can be written as the Stieltjes
spectrum (2)
ïntegral (3)
with B(E) =
1@i@E)i2
(4)
the spectral density function and I’(e) the associated distribution 1221. An approximation to the integral of eq. (3) is obtaïned from the set of L2 functions &, i = 1 ,n, which diagonalize j!?, dïstinct from the COntinUUm functions (p, of eq. (2), in the form n s
(-9 wherc Q = l(@lQ>i2, 610
and the Fi are the eigenvalues of Hin the set & Aithough the approximation of eq. (5) does not exhibit the correct ìmaginary discontinuity of eq. (3) across the real axis, the concepts of Stieltjes and Tchebycheff derivatives [ 13-19 J can be employed to obtah approximations to the spectra1 density g(e) of eq. (4). Alternatively, vahìes of eq. (5) for complex z can be continued to the real axis employing appropriate analytic farms [14], and for particular one-electron hamiltonians and basis functions, alpproximations to g(e) are obtained from the 5, ri and known equivalent quadratures CS---LO] . Moreover, when the coordinates of the hamihonian appearíng in eq. (1) are rotated to complex values, the imaginary discontinuity of eq. (3) formerly across the real atis appears along a ray in the lower haif plane, and an approxïmation to the spectral density is obtained directly from eq. (5) and fhe now complex < and Fj values [11,12] . It is 3ko of interest to investigate the melhods of reference densities and polynomïal expansions [20,21] in obtaining approximations to the spectral density g(c) from the variationahy calculated 5 and ?i values.
with respect to some
= (Wz-H)-ll@),
where the hamiltonian
1 August 1976
CEIELCICALPHYSICS LETTERS
(6)
3. Reference
densities and polynomíal
expansions
We consider an arbitrary but known reference density W(X) detììed on [c, bj , which is not necessarily a classica1 density [23 1, and an arbitrary but known mapping T(x) of [O,m] onto [a, b] . The differential of the distrïbution in eq. (3) takes the form dr(c)
= g(e) de = g [ZQ)]
T’(x) dx
(7)
under the mapping E = T(X). The density on the righthand side of eq. (7) can always be approximated by the expression [20,21) +
f The convergente of eq. (8) must be established in each case, in which connection it is important to chose the reference decsity tisely. For erample, the choke W(x) = 1, ZW = x. and the use of Legepdre polynomials is satisfactory for the spectral density of normal mode vïïrations of a monatomic chaïn, originaliy investïgated by Montroll /20], but is kapproprïate for the dïtonüc chaïn, due to the presence of van Have tingularities in the density. In the case of electronscattering and photoiotiation calctitions, for which there
are no sin@srities in the spectrai density, it is important to incorporate the E -+ 0 and - behaviors correctly in W(x).
N
gwm~)l~‘(x) = we c c,4,(x), where the Q&) are polynomials respect to W(x),
orthonormal
(8)
Q#)qp)
Wx)
dx = bij>
[23] , eq. (12) can be written in
(13) where (9)
and the ci are appropriate weighting coefficients. The latter are obtained from eq. (8) and the orthonormality relations of eq. (9) in the form b ci =
sical density W,(x) the form
with
b s a
1 August 1976
CHEMICAL PHYSICS LETTERS
Volume 4 1, number 3
s 4i(xkIT(x)l “tc) dX a
de. (10) s- q&_+(E)]g(E) 0 Evidently , the coefficients of eq. (10) are polynomial or more genera1 moments of the unknown spectral density g(e), depending on the nature of the mapping =
W). Convergent approximations to the mzments of eq. (10) are obtamed fsom the L2 basis set $, i = 1, n, which diagonalizes the many-electron hamïltonïan of eqs. (1) and (2), in the farm*
(11) where the Fj are given by eq. (6) and E;- = x(sj) is the associated jth eigenvahre of H in the basis ~ii- Introducing eq. (11) ïnto eq. (8) gives
is the Christoffel-Darboux function [24] associated with the classical density W,.(X). Under further special conditions eq. (13) becomes the equivalent-quadrature approximation of Reinhardt and ca-werkers [S-71. To demonstrate this, note that when H is a single-particle hamïltonïan and the S are chosen to be one of the special set of functions (Laguerre or oscillator) for which an equivalent quadrature is known [S-71, eq. (13) takes the form
2 ri
p)(e) --rr(x)i=l w(x)
K
N
(x x-) ’ J’
where w(x) and f(x) refer to appropriate equivaientquadrature densities and mappïngs, respectively, and the xi are the associated classical quadrature abscissa. Eq. (15) is a generalization of the equivalent-quadrature result ES-73 to an arbitrary energy e*. Wiien eq. (15) is evaluated at an energy point ei = t(Xi) associated with a classical abscissa Xi, and the number of terms N in the expansion of eq. (14) is set equal to the number of basis functions n employed, we obtam
(12) as an approximate spectral density. The density of eq. (12) is a completely general approximation to g(e) in that thereference density W(x), the mappíng qx), and the L2 basis set $ are & arbitrary. When W(x) is chosen to be a known clas* ït is, of course, necessary that Z’(X)be of such form that the integrak of eq. (10) converge. In this case, the convergence of square-irltegrable approximations [eq. (i 1)] to modïfïed moments can be investigated and established in each individu&case. We dehte an explicit label n, which is generally distinct from the number of terms iV employed in the polynomial expansion [eq. (8)) , on g(N)(e) in the foUowing, and, rather, employ the notation zw to denote a spectral density obtained from variationally determined spectral moments.
the equivalentquadrature result of Reinhardt and cowerkers*, where Wi is a classical quadrature weight and use bas been made in eq. (16) of the expression [24] K,r(xi, xj) = ( ‘l”i)‘jj.
(17)
* Appropriate polynomial interpolation can also be employed in conjunction with the equivalentquadrature approach in order to obtain an approximate speetral density over a continuous range of energy [ 14]_ * That eq. (16) can converge to g(e) in the limit n + rr~is clear from the present moment-theory development, and is related to the fact that in the equivalentquadrature approach the functions 3 i = 1,n. or more generally the functions biorthogonal to this set, can fully span the space of the test vector Q>of eq. (1) in the limit II -F -.
611
4. itiustrative
1 August 1976
CHEMICAL PHYSICS LETTERS
Völmle 41, number 3
of eqs. (8) to (15) by choosing g(e) = A(~)/f’(i), which case eq. (15) gives
applications
The Fredholm approach to s-wave scattering of ekctrons from atomic hydrogen in the static approxïmation 1191 provides a useful example for studying the corívergence of the polynomïal expansion of eq. (15). The appropriate dispersion relation for the swave Fredholm determinant is
&)(e)
= (44/h2) (1 - X2)I’2 n
N
x lz
2
qC1
-xj124f(xi)4~(x)
= (4,X2)(1-x2)1/2I$
(18) where A(c) is the unknow-n spectral density. Diagonalïzation of H(o) and H = XZ(O)+ Vin a Laguerre basis 11-31 xi = hre-fQj_ generatcs
@),
i = 1,2, . ..> n
(19)
in
q(l
-x~)~K~(x,x~).
(24)
Eq. (24) reduces to the equivalent-quadrature result of eq. (23) when x = xi. Approximations to the ima@nary and real parts of the Fredholm determinant Im@)(e)
= -7~4(~)(~) >
GW
the approximation
and to the phase shift where the Ei and ri are obtaïned dirëctly from the eigenvalues of H(o) and Hin the basis of eq. (19). Eq. (20) is equivalent to a Tchebycheff quacirature with r1-31 w(x) = (1 -x2)=,
Vla)
xi = cos[i7T/(n + 1)] ,
(21bj
wi = [7T/(n + l)] sir? [in/(rz + l)] ,
cw
under the mapping E =
t(x) = i
A2(1 +x)/(
1 -x)_
(22)
Consequently, the equivalent-quadrature approximation to ‘rhe spectra1 density A(e) in this case is obtained from eq. (16) with g(c) s A(E) in the form
&)(e_)1 =
4ry?f + 1) {1 _ cos [zYf/(n+ 1)] }2 x%r
sin [k/(n + l)]
’
(23)
A generalization of eq. (23) to an arbitrary energy point e is obtained from eq. (15) employing the appropriate recurrence relation to evaluate the required (Tchebycheff3 orthonormal polynomials 1241. Since ihe area under the density A(E) is unbounded [25] , its power moments do not converge, and the polynomid expansion of eqs. (8) to (15) is unstable when the identifïcation g(e) = A(e) is employed. Rapid convergence, however, is obtained from the deveiopment 612
6@)(e)
= tan-’
[-im
D(‘)(e)/Re
D(“)(e)]
,
Wc)
are obtained from the development of eqk. (18) to (24) empioying up to n = 50 basis functions [eq. (lg)] wirh h = 3.75. The polynomial expansion of eq. (24) is found to be rapiclly convergent, with N 2 i 0 providing highly satisfactory values. In fig. 1 are shown the (convergent) results of eqs. (23), (24) and (25) for n = 30 and N = 10, obtained employing a disper. sion correction technique to aid in evaluating the principal vake ïntegral of eq. (25b). Evidentiy eqs. (24) and (25a) provide the correct interpolation of the equivalentquadrature values, ensuring that the phase shift obtained from eqs. (24) and (25) is in good agreement with the correct results [ 191. Photoionization in the negative hydrogen ion provides a suitable example for investigating the convergence of eq. (12), for which there is no corresponding equivalentquadrature development. In this case the test vector @ appearing in eqs. (1) to (6) is given by r_~@~,where I-(is the dipole moment operator and . oo the ground state eigenfunction, and g(e) corresponds to the squared transition rnomgnt density. We employ the rnapping E = nx)
= ‘JX,
where et (= 0.75 eV) is the photoionization old, and the reference density
(26) thresh-
Volumt 41, number 3
CHENWAL
3 ö
_*5
t
OOI
8
,ttd
t
1
rrttrtl
0.1
f t ltrrttl 10
I
ENERGY
100
- E (o LI.)
xsy
10
1 IIII
FREQUENCY
Fig. 1. Fredholm determinant and associated phasc shift for the s-wave elastic scattering of electrons from Is atomic hydrogen in the static approximation 1191; o equïvatentqnadrature results of eqs. (23) and f25a) for the imagínary part of the Fredholm determinant empfoying n = 3ö; polynomial expansion for the reaband imaginary parts of the Fredholm determinant of eqs. (243, (25a), and (25b) employing n=30andN=lO;--phase shift of cqs. (24) and (25~) employing ri = 30 and N = 10.
ív(x) = (4/3ne,) (1 - xp
1 August 1976
PHYSICS LETTERS
(27)
which is the familiar Bethe-Ohmura approximation to the squared transition moment profIle in H- [ 143 . Since the density of eq_ (27) is of the Jacobi type, the polynomïak qi (x) appearíng in eq. (12) are easîly 2vaL uated employìng the known recurrence relation [23]. Previously reported pseudo-spectra for H- [lS] are employed in evaluating the moments of eq. (1 l), which =e linear combinations of the associated convergent power moments. Eqs. (26) and (271, and the associated Jacobi polynoxnials and variationally determined polynomial moments, are employed in the evaluation of eq. (12) for N G 22. The resulting photoionìzation profiles are fotmd to converge to the experimental [26] and prevîous theoretical f27 J values for N e 10, the N = 15 and 16 profties are mutually indistinguishable, indicating convergente of the polynomial expansion,
-
c (a
LI.)
Fig. 2. Photoionization
profile in the negative hydrogen ion; --polynomial expansion of cqs. (12), (26), and (27) employing variaticmalíy determined moments fis] for N = 9; for N = X5,16: 6 experimental data g26] normalïzed to thc N = 15 results at E = 0.08629 au; A prevïous theoretical calculations [ 271.
and for N > 18 small high frequency osciflations are present in the profUes_ In fig. 2 are shown the N = 9 and N = 15, i6 photoionïzation profties, as well as experimental 1263 and previous theoretical 1271 values. Evidently, the polynomialexpansïon results are in good accord with the experimental and previous theoretical values, the smrttl oscillatory structures present in the former attributable to the use of variationally determined spectral moments. Somewhat smoother profì’ies can be obtained from the poiynomial-expansion approach employing appropriate averaging procedures.
5. Concludïng remarks
.
Pclynomial expansions and modifïed moments are employed ìn imaging the spectra2 densities that ark in electron-scattering and photoio~zatïon cafculations. Square integrable basis functions and Ritz variational calcnlations are employed in calculations of 613
Volume 41, number 3
CHEMICAL PHYSICS LETTERS
tke required spectral moments. An ihstrative apq plication ta the Freáholm determhant for s-wave scattering of electrons by IS hydrogen in the static approximation indicates that the method is a useful generalization of previousy reported equivalentquadrature techniques [ 141. Rapidly convergent photoionization profiles in good accord with experimental [2S] and previous theoretical 1271 results are obtained in the case of the negative hydrogen ion. These results suggest that the polynomial-expansion technique should provide a useful complement to previously described approaches [ 1-191 for scatterïng and photoionization calculations in Hilbert space,
Achowledgement It is a pleasure for P.W.L. to thank F. Weïnhold for his generous support at Stanford University, and to thank W.P. Reinhardt and tte JILA Fellows for theïr kind hospitality.
References
i11 T.S.
hiurtaugh and W.P. Reinhardt, Chem. Phys. Letters 11 (1972) 562. 121W.P. Reinhardt, D.W. Oxtoby and T.N. Rescigno, Phys. Rev. Letters 28 (1972) 401. :31 T.S. Murtaugh and W.P. Reinhardt, 3. Chem. Phys. 57 (1972) 2129; 59 (1973) 4900. 141 J.T. Broad acd W.P. Reinhardt, J. Chem. Phys. 60 (1974) 2182. [SI E.J. HeIIer, W.P. Reinhardt and H.A. Yamani, J. Comp. Phys. 13 &73)_536. [61 E.J. Heller, T.N. Rescigno and W.P. Reinhardt, Phys. Rev. A 8 (1973) 2946.
614
1 Augïïst 1976
[7] H.A. Yamani and W.P. Reinhardt, Phys. Rev. A 11 (1975) 1144. [B] E.J. Heller and H.A. Yamanï, Phys. Rev. A 9 (1974) 1201. [9] E.d. HeIIer and H.A. Yamani, Phys. Rev. A9 (1974) 1209. [lO] E.J. Heller, Phys. Rev. A 12 (1975) 1222. [ 111 T.N. Rescigno, C.W. McCurdy and V. hfclcoy, Phys. Rev. A 9 (1974) 2409. 1121 T.N. Rescîgno, C.W. McCurdy and V. hfcby, J. ChemPhys. 64 (1976) 477. [ 131 P.W. Lxnaoff, Chem. Phys. Letters 22 (1973) 60. t141P.W. Langhoff and C.T. Corcoran, J. Chem. Phys. 61 (1974) 146. 1151P.W. Langhoff, J.S. Sims and C.T. Corcoran, Phys. Rev. A 10 (1974) 829. 1161 P.W. Langhoff, Intern. J. Quantum Chem. 8s (1974) 347. 1171 P.W. Langhoff and C.T. Corcoran, Chem. Phys. Letters 40 (1976) 361. 1181P.W. Langhoff, C.T. Corcoran, J.S. Sims, F. Weinhold and R.M. Giover, Phys. Rev. A (1976), to be published. 1191P.W. Langhoff and W.P. Reinhardt, Chem. Phys. Letters 24 (1974) 495. 1201E.W. MOII~IGJ~,J. Chem. Phys. 10 (1942) 218; 11 (1943) 481. 1211J.C. Wheeler, M.G. Prais and C. Blumstein, Phys. Rev. B 10 (1974) 2429. 1271J.A. Shohat and J.D. Tamarkin, The problem of mcments (American Mathematical Socxety, Providence, 1943). 1231 U.W. Hochstrasser, in: Handiiook of mathematica1 functions, eds. M. Abramowitz and LA. Stegun (US Govt. Printiug Office, Washington, 1964) cb. 22. [24] G. Szego, Orthogonal polynomïals (American hfathematicaI Society, Providence, 1959). 1251 L.I. Schiff, Quantum mechanics (hfcGraw-HiII, New York, 1968) p. 349. 1261 S.J. Smith and D.S. Burch, Phys. Rev. 116 (1959) 1125. 1271 N.A. Doughty, P.A. Fraser and R.P. McEachran, hlonthly Notxes Roy. Astron. Sec. 132 (1966) 255.
1 August 1976
CHEMICAL PHYSICS LETTERS
Volume 41, number 3
EXPANSIONS
FOR SPECTRAL
DENSlTLES
C.T. CORCORAN* Department of Ci’zemistry. Indiana University, Bioomìngron,
Indiana 47401, USA
and
P.W. LAXGHOFF** Department o_fChemistry, Indiana University, Bkwmington, Indìana 47401, USA and Department of Chemìstry, Stanford Universìty, Stanford, Gzlifornia 94305. USA and Joint Institute for Laboratory Astrophysìcs, Natìonal Bureau of Standardsand Universìty of Glorado. Boulder, Colorado 80309, USA Received 12 April 1976
Modified moments and polynomial expansions are employed in imaging scattering and photoionïzation
the spectra1 densities that arise in electron-
calculations.
1. lntroduction
rotation techniques in the latter connection. me theory of moments and Stieltjesintegration have al-
electron-scattering cross sections [lV] . Altllough the various approaches reported are satísfactory*, generalïzations and alternative perspectíves are always useful. In the present letter, modifïed moments and polynomíal expansions [20,21] are employed in approximating the spectral densítíes that aríse ín electronscattering and photoionization calculations. A general technique ís thereby provided for performing scattering and photoionization calculations in Hilbert space for an arbítrary many-electron system, employíng arbitrary basis sets of L2 functions. In the case of appropriate oneelectron hamiltonians and Laguerre or oscillator basis functions, the technique gives a generalization of the previously reported equivalentquaclrature approach [5-71, providing an approximate, convergent spectra1 densíty over a continuous range of energies. Moreover, when the density is eval-
so been employed in L2 calculations of photoabsorption and díspersion profdes [ 13-1 S] and elastic
* A consïderable number of publications dealing wïth the use
Considerable attention has focused recently on calculations of electron-scattering and photoionization cross sections entirely in Hilbert space. Reinhardt and co-workers have reported Lz calcuìatíons of Fredholm determinants El-31 and dípole polarizabilitíes [4] at complex energies and their analytic continuation to the real axis, and more recently have described an L2 equivalent-quadrature technique from which phase shifts and photoionization cross sections are obtained directly [S-71. In a separate but relatec! development, Heller and Yamani [8-101 have described a J-matrix approach to scattering and photoionization calculatíons, and McKoy and cowerkers [ 11,123 have reported the use of coordinate-
* Proctor and Gamble Fellow. ** On leave of absente from Indiana University, formerly visiting Stanford Unïversity, and presently JILA Visiting Fellow.
of square-integsable basis functions in scattering and other continuum calculations has appeared recently. The develop ment reported in the present letter is perhaps most closely related to the pubkations clterl in refs. [l-191, and no attempt is mzde hese at a complete citation of the recent literature in this area.
609
Volume 41, number 3
uated at the appropriate quadrature abscissa, results identical to the equivaIentquadrature technique are obtained, indicating that convergente of the latter approach can be estabhshed employing the rheory of moments. Polynomial expansions for the spectral densities that arise in scattering and photoionization calculations are descrïbed in sections 2 and 3, and connections with the equivalentquadrature technique are ìndicated. fllustrative apphcaiions Li the cases of Tlectron-hydrogen elastic scatterin& and photdonization in the negative hydrogep ion are gïven in section 4. The rapid convergente obtained in these exarrp!es suggests that the polynomial-expansion approach for imaging spectra1 densities should provide a useful addition tc the growing list of techniques for performing scattering and photoionization calculations entirely in Hilbert space. Some concluding tema& are gíven in sectiori 5.
2. Spectra1 densities The mean vaIue of a resolvent test function a, I$(z)
(W-
e)$c =o,
(1) Hhas
a continuous
O
can be written as the Stieltjes
spectrum (2)
ïntegral (3)
with B(E) =
1@i@E)i2
(4)
the spectral density function and I’(e) the associated distribution 1221. An approximation to the integral of eq. (3) is obtaïned from the set of L2 functions &, i = 1 ,n, which diagonalize j!?, dïstinct from the COntinUUm functions (p, of eq. (2), in the form n s
(-9 wherc Q = l(@lQ>i2, 610
and the Fi are the eigenvalues of Hin the set & Aithough the approximation of eq. (5) does not exhibit the correct ìmaginary discontinuity of eq. (3) across the real axis, the concepts of Stieltjes and Tchebycheff derivatives [ 13-19 J can be employed to obtah approximations to the spectra1 density g(e) of eq. (4). Alternatively, vahìes of eq. (5) for complex z can be continued to the real axis employing appropriate analytic farms [14], and for particular one-electron hamiltonians and basis functions, alpproximations to g(e) are obtained from the 5, ri and known equivalent quadratures CS---LO] . Moreover, when the coordinates of the hamihonian appearíng in eq. (1) are rotated to complex values, the imaginary discontinuity of eq. (3) formerly across the real atis appears along a ray in the lower haif plane, and an approxïmation to the spectral density is obtained directly from eq. (5) and fhe now complex < and Fj values [11,12] . It is 3ko of interest to investigate the melhods of reference densities and polynomïal expansions [20,21] in obtaining approximations to the spectral density g(c) from the variationahy calculated 5 and ?i values.
with respect to some
= (Wz-H)-ll@),
where the hamiltonian
1 August 1976
CEIELCICALPHYSICS LETTERS
(6)
3. Reference
densities and polynomíal
expansions
We consider an arbitrary but known reference density W(X) detììed on [c, bj , which is not necessarily a classica1 density [23 1, and an arbitrary but known mapping T(x) of [O,m] onto [a, b] . The differential of the distrïbution in eq. (3) takes the form dr(c)
= g(e) de = g [ZQ)]
T’(x) dx
(7)
under the mapping E = T(X). The density on the righthand side of eq. (7) can always be approximated by the expression [20,21) +
f The convergente of eq. (8) must be established in each case, in which connection it is important to chose the reference decsity tisely. For erample, the choke W(x) = 1, ZW = x. and the use of Legepdre polynomials is satisfactory for the spectral density of normal mode vïïrations of a monatomic chaïn, originaliy investïgated by Montroll /20], but is kapproprïate for the dïtonüc chaïn, due to the presence of van Have tingularities in the density. In the case of electronscattering and photoiotiation calctitions, for which there
are no sin@srities in the spectrai density, it is important to incorporate the E -+ 0 and - behaviors correctly in W(x).
N
gwm~)l~‘(x) = we c c,4,(x), where the Q&) are polynomials respect to W(x),
orthonormal
(8)
Q#)qp)
Wx)
dx = bij>
[23] , eq. (12) can be written in
(13) where (9)
and the ci are appropriate weighting coefficients. The latter are obtained from eq. (8) and the orthonormality relations of eq. (9) in the form b ci =
sical density W,(x) the form
with
b s a
1 August 1976
CHEMICAL PHYSICS LETTERS
Volume 4 1, number 3
s 4i(xkIT(x)l “tc) dX a
de. (10) s- q&_+(E)]g(E) 0 Evidently , the coefficients of eq. (10) are polynomial or more genera1 moments of the unknown spectral density g(e), depending on the nature of the mapping =
W). Convergent approximations to the mzments of eq. (10) are obtamed fsom the L2 basis set $, i = 1, n, which diagonalizes the many-electron hamïltonïan of eqs. (1) and (2), in the farm*
(11) where the Fj are given by eq. (6) and E;- = x(sj) is the associated jth eigenvahre of H in the basis ~ii- Introducing eq. (11) ïnto eq. (8) gives
is the Christoffel-Darboux function [24] associated with the classical density W,.(X). Under further special conditions eq. (13) becomes the equivalent-quadrature approximation of Reinhardt and ca-werkers [S-71. To demonstrate this, note that when H is a single-particle hamïltonïan and the S are chosen to be one of the special set of functions (Laguerre or oscillator) for which an equivalent quadrature is known [S-71, eq. (13) takes the form
2 ri
p)(e) --rr(x)i=l w(x)
K
N
(x x-) ’ J’
where w(x) and f(x) refer to appropriate equivaientquadrature densities and mappïngs, respectively, and the xi are the associated classical quadrature abscissa. Eq. (15) is a generalization of the equivalent-quadrature result ES-73 to an arbitrary energy e*. Wiien eq. (15) is evaluated at an energy point ei = t(Xi) associated with a classical abscissa Xi, and the number of terms N in the expansion of eq. (14) is set equal to the number of basis functions n employed, we obtam
(12) as an approximate spectral density. The density of eq. (12) is a completely general approximation to g(e) in that thereference density W(x), the mappíng qx), and the L2 basis set $ are & arbitrary. When W(x) is chosen to be a known clas* ït is, of course, necessary that Z’(X)be of such form that the integrak of eq. (10) converge. In this case, the convergence of square-irltegrable approximations [eq. (i 1)] to modïfïed moments can be investigated and established in each individu&case. We dehte an explicit label n, which is generally distinct from the number of terms iV employed in the polynomial expansion [eq. (8)) , on g(N)(e) in the foUowing, and, rather, employ the notation zw to denote a spectral density obtained from variationally determined spectral moments.
the equivalentquadrature result of Reinhardt and cowerkers*, where Wi is a classical quadrature weight and use bas been made in eq. (16) of the expression [24] K,r(xi, xj) = ( ‘l”i)‘jj.
(17)
* Appropriate polynomial interpolation can also be employed in conjunction with the equivalentquadrature approach in order to obtain an approximate speetral density over a continuous range of energy [ 14]_ * That eq. (16) can converge to g(e) in the limit n + rr~is clear from the present moment-theory development, and is related to the fact that in the equivalentquadrature approach the functions 3 i = 1,n. or more generally the functions biorthogonal to this set, can fully span the space of the test vector Q>of eq. (1) in the limit II -F -.
611
4. itiustrative
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Völmle 41, number 3
of eqs. (8) to (15) by choosing g(e) = A(~)/f’(i), which case eq. (15) gives
applications
The Fredholm approach to s-wave scattering of ekctrons from atomic hydrogen in the static approxïmation 1191 provides a useful example for studying the corívergence of the polynomïal expansion of eq. (15). The appropriate dispersion relation for the swave Fredholm determinant is
&)(e)
= (44/h2) (1 - X2)I’2 n
N
x lz
2
qC1
-xj124f(xi)4~(x)
= (4,X2)(1-x2)1/2I$
(18) where A(c) is the unknow-n spectral density. Diagonalïzation of H(o) and H = XZ(O)+ Vin a Laguerre basis 11-31 xi = hre-fQj_ generatcs
@),
i = 1,2, . ..> n
(19)
in
q(l
-x~)~K~(x,x~).
(24)
Eq. (24) reduces to the equivalent-quadrature result of eq. (23) when x = xi. Approximations to the ima@nary and real parts of the Fredholm determinant Im@)(e)
= -7~4(~)(~) >
GW
the approximation
and to the phase shift where the Ei and ri are obtaïned dirëctly from the eigenvalues of H(o) and Hin the basis of eq. (19). Eq. (20) is equivalent to a Tchebycheff quacirature with r1-31 w(x) = (1 -x2)=,
Vla)
xi = cos[i7T/(n + 1)] ,
(21bj
wi = [7T/(n + l)] sir? [in/(rz + l)] ,
cw
under the mapping E =
t(x) = i
A2(1 +x)/(
1 -x)_
(22)
Consequently, the equivalent-quadrature approximation to ‘rhe spectra1 density A(e) in this case is obtained from eq. (16) with g(c) s A(E) in the form
&)(e_)1 =
4ry?f + 1) {1 _ cos [zYf/(n+ 1)] }2 x%r
sin [k/(n + l)]
’
(23)
A generalization of eq. (23) to an arbitrary energy point e is obtained from eq. (15) employing the appropriate recurrence relation to evaluate the required (Tchebycheff3 orthonormal polynomials 1241. Since ihe area under the density A(E) is unbounded [25] , its power moments do not converge, and the polynomid expansion of eqs. (8) to (15) is unstable when the identifïcation g(e) = A(e) is employed. Rapid convergence, however, is obtained from the deveiopment 612
6@)(e)
= tan-’
[-im
D(‘)(e)/Re
D(“)(e)]
,
Wc)
are obtained from the development of eqk. (18) to (24) empioying up to n = 50 basis functions [eq. (lg)] wirh h = 3.75. The polynomial expansion of eq. (24) is found to be rapiclly convergent, with N 2 i 0 providing highly satisfactory values. In fig. 1 are shown the (convergent) results of eqs. (23), (24) and (25) for n = 30 and N = 10, obtained employing a disper. sion correction technique to aid in evaluating the principal vake ïntegral of eq. (25b). Evidentiy eqs. (24) and (25a) provide the correct interpolation of the equivalentquadrature values, ensuring that the phase shift obtained from eqs. (24) and (25) is in good agreement with the correct results [ 191. Photoionization in the negative hydrogen ion provides a suitable example for investigating the convergence of eq. (12), for which there is no corresponding equivalentquadrature development. In this case the test vector @ appearing in eqs. (1) to (6) is given by r_~@~,where I-(is the dipole moment operator and . oo the ground state eigenfunction, and g(e) corresponds to the squared transition rnomgnt density. We employ the rnapping E = nx)
= ‘JX,
where et (= 0.75 eV) is the photoionization old, and the reference density
(26) thresh-
Volumt 41, number 3
CHENWAL
3 ö
_*5
t
OOI
8
,ttd
t
1
rrttrtl
0.1
f t ltrrttl 10
I
ENERGY
100
- E (o LI.)
xsy
10
1 IIII
FREQUENCY
Fig. 1. Fredholm determinant and associated phasc shift for the s-wave elastic scattering of electrons from Is atomic hydrogen in the static approximation 1191; o equïvatentqnadrature results of eqs. (23) and f25a) for the imagínary part of the Fredholm determinant empfoying n = 3ö; polynomial expansion for the reaband imaginary parts of the Fredholm determinant of eqs. (243, (25a), and (25b) employing n=30andN=lO;--phase shift of cqs. (24) and (25~) employing ri = 30 and N = 10.
ív(x) = (4/3ne,) (1 - xp
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PHYSICS LETTERS
(27)
which is the familiar Bethe-Ohmura approximation to the squared transition moment profIle in H- [ 143 . Since the density of eq_ (27) is of the Jacobi type, the polynomïak qi (x) appearíng in eq. (12) are easîly 2vaL uated employìng the known recurrence relation [23]. Previously reported pseudo-spectra for H- [lS] are employed in evaluating the moments of eq. (1 l), which =e linear combinations of the associated convergent power moments. Eqs. (26) and (271, and the associated Jacobi polynoxnials and variationally determined polynomial moments, are employed in the evaluation of eq. (12) for N G 22. The resulting photoionìzation profiles are fotmd to converge to the experimental [26] and prevîous theoretical f27 J values for N e 10, the N = 15 and 16 profties are mutually indistinguishable, indicating convergente of the polynomial expansion,
-
c (a
LI.)
Fig. 2. Photoionization
profile in the negative hydrogen ion; --polynomial expansion of cqs. (12), (26), and (27) employing variaticmalíy determined moments fis] for N = 9; for N = X5,16: 6 experimental data g26] normalïzed to thc N = 15 results at E = 0.08629 au; A prevïous theoretical calculations [ 271.
and for N > 18 small high frequency osciflations are present in the profUes_ In fig. 2 are shown the N = 9 and N = 15, i6 photoionïzation profties, as well as experimental 1263 and previous theoretical 1271 values. Evidently, the polynomialexpansïon results are in good accord with the experimental and previous theoretical values, the smrttl oscillatory structures present in the former attributable to the use of variationally determined spectral moments. Somewhat smoother profì’ies can be obtained from the poiynomial-expansion approach employing appropriate averaging procedures.
5. Concludïng remarks
.
Pclynomial expansions and modifïed moments are employed ìn imaging the spectra2 densities that ark in electron-scattering and photoio~zatïon cafculations. Square integrable basis functions and Ritz variational calcnlations are employed in calculations of 613
Volume 41, number 3
CHEMICAL PHYSICS LETTERS
tke required spectral moments. An ihstrative apq plication ta the Freáholm determhant for s-wave scattering of electrons by IS hydrogen in the static approximation indicates that the method is a useful generalization of previousy reported equivalentquadrature techniques [ 141. Rapidly convergent photoionization profiles in good accord with experimental [2S] and previous theoretical 1271 results are obtained in the case of the negative hydrogen ion. These results suggest that the polynomial-expansion technique should provide a useful complement to previously described approaches [ 1-191 for scatterïng and photoionization calculations in Hilbert space,
Achowledgement It is a pleasure for P.W.L. to thank F. Weïnhold for his generous support at Stanford University, and to thank W.P. Reinhardt and tte JILA Fellows for theïr kind hospitality.
References
i11 T.S.
hiurtaugh and W.P. Reinhardt, Chem. Phys. Letters 11 (1972) 562. 121W.P. Reinhardt, D.W. Oxtoby and T.N. Rescigno, Phys. Rev. Letters 28 (1972) 401. :31 T.S. Murtaugh and W.P. Reinhardt, 3. Chem. Phys. 57 (1972) 2129; 59 (1973) 4900. 141 J.T. Broad acd W.P. Reinhardt, J. Chem. Phys. 60 (1974) 2182. [SI E.J. HeIIer, W.P. Reinhardt and H.A. Yamani, J. Comp. Phys. 13 &73)_536. [61 E.J. Heller, T.N. Rescigno and W.P. Reinhardt, Phys. Rev. A 8 (1973) 2946.
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[7] H.A. Yamani and W.P. Reinhardt, Phys. Rev. A 11 (1975) 1144. [B] E.J. Heller and H.A. Yamanï, Phys. Rev. A 9 (1974) 1201. [9] E.d. HeIIer and H.A. Yamani, Phys. Rev. A9 (1974) 1209. [lO] E.J. Heller, Phys. Rev. A 12 (1975) 1222. [ 111 T.N. Rescigno, C.W. McCurdy and V. hfclcoy, Phys. Rev. A 9 (1974) 2409. 1121 T.N. Rescîgno, C.W. McCurdy and V. hfcby, J. ChemPhys. 64 (1976) 477. [ 131 P.W. Lxnaoff, Chem. Phys. Letters 22 (1973) 60. t141P.W. Langhoff and C.T. Corcoran, J. Chem. Phys. 61 (1974) 146. 1151P.W. Langhoff, J.S. Sims and C.T. Corcoran, Phys. Rev. A 10 (1974) 829. 1161 P.W. Langhoff, Intern. J. Quantum Chem. 8s (1974) 347. 1171 P.W. Langhoff and C.T. Corcoran, Chem. Phys. Letters 40 (1976) 361. 1181P.W. Langhoff, C.T. Corcoran, J.S. Sims, F. Weinhold and R.M. Giover, Phys. Rev. A (1976), to be published. 1191P.W. Langhoff and W.P. Reinhardt, Chem. Phys. Letters 24 (1974) 495. 1201E.W. MOII~IGJ~,J. Chem. Phys. 10 (1942) 218; 11 (1943) 481. 1211J.C. Wheeler, M.G. Prais and C. Blumstein, Phys. Rev. B 10 (1974) 2429. 1271J.A. Shohat and J.D. Tamarkin, The problem of mcments (American Mathematical Socxety, Providence, 1943). 1231 U.W. Hochstrasser, in: Handiiook of mathematica1 functions, eds. M. Abramowitz and LA. Stegun (US Govt. Printiug Office, Washington, 1964) cb. 22. [24] G. Szego, Orthogonal polynomïals (American hfathematicaI Society, Providence, 1959). 1251 L.I. Schiff, Quantum mechanics (hfcGraw-HiII, New York, 1968) p. 349. 1261 S.J. Smith and D.S. Burch, Phys. Rev. 116 (1959) 1125. 1271 N.A. Doughty, P.A. Fraser and R.P. McEachran, hlonthly Notxes Roy. Astron. Sec. 132 (1966) 255.