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Variational calculations of rovibrational states of Li2K+
Chemical Physics 172 (1993) 247-258 North-Holland
Variational calculations of rovibrational states of Li,K+ F. Wang and E.I. von Nagy-Felsobuki
’
Department of Chemistry, The University of Newcastle, Callaghan. NSW 2308, Australia Received 19 October 1992
The low-lying rovibrational states for the ground electronic state of Li*K+ were calculated using an ab initio variational solution of the nuclear Schriidinger equation. A discrete configuration interaction potential energy surface was generated and an analytical representation was obtained using a power series expansion. This force field was embedded in the Eckart-Watson Hamiltonian from which rovibrational wavefunctions and eigenenergies were variationally calculated. An SCF dipole moment surface was generated and used to calculate absolute line intensities and square dipole moment matrix elements for some of the most intense transitions within the P, Q and R branches, between the vibrational ground state and the low-lying rovibrational states.
1. Introduction Theoretical studies on alkali cluster ions have centered on their electronic structure, stability and equilibrium geometry [ l-31. While most work has focussed on the homogeneous cluster ions [ l-3 1, more recent studies have been extended to the electronic structure of mixed clusters of form X,,Y + (X, Y = Li, Na, K) based on pseudopotential [ 4,5 ] and all-electron [ 1,6-81 molecular orbital methodologies. This has been partly prompted by the observations of the heterogeneous alkali cluster ions in supersanic expansions [ 9,101. So far no observatio?phave been made on their rovibrational s$_c&r& even though such information would be valuable with respect to their proposed significance in a number of technological processes [ 111. Recently, ab initio solutions of the “complete” molecular Schriidinger equation have considerably unravelled the rovibrational structure of small molecules [ 6-8,12-22 1. By “complete” we mean that an ab initio Born-Oppenheimer potential energy (PE) surface was generated and embedded in the nuclear Schriidinger equation in order to variationally yield rovibrational eigenvalues and eigenfunctions [ 12 1. In the most celebrated exthis regard, H: is perhaps ample, since quanta1 investigations correctly pre-
’ To whom correspondence should be addressed. 0301-0104/93/$06.00
dieted its infrared spectrum near the potential energy minimum prior to experimental observations [ 1217]. Due to lack of computing capacity, more speculative configuration interaction (CI) surfaces are usually entertained for electron-dense molecules. Simple CI truncations, such as single and double substitutions from a single Hartree-Fock reference determinant (SDCI) and basis set limitations are imposed in order to make computations tractable [6-8,18221. For molecules containing atoms beyond the first row, additional restrictions are imposed. For example, in the case of L&Na+, LiNa$, KLiNa+ and K,Li+ von Nagy-Felsobuki and co-workers [ 6-8,182 1] have restricted the CI subspace to only valence molecular orbitals (coined the “frozen-core” approximation and labelled FC ) . Alternatively, Pavolini and Spiegelmann [ 5 ] in their study of X2Y+ (X, Y = Li, Na, K) clusters, have resorted to a pseudopotential CI methodology, because of the need for a similar trade-off. Whatever the limitations, the motivation for these studies is to produce PE surfaces which are not only cost-effective, but spectroscopically predictive. Even for triatomic molecules, Eckart’s notion of an embedded equilibrium geometry may become problematic, since there are different rovibrational Hamiltonians for bent and linear nuclear configurations (as derived by Watson [23,24] ). A smooth transi-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
248
F. Wang and E.I. von Nagy-Felsobuki /Chemrcal Physics 172 (1993) 247-258
tion cannot be made between the two Hamiltonians, because singularities occur in the mass-dependent potential energy operator (termed the Watson operator) of the bent Hamiltonian for linear configurations. Consequently, Tennyson and Sutcliffe [ 25,261 have developed a rovibrational Hamiltonian in terms of a body-fixed scattering coordinate system, in which there is no embedded equilibrium geometry. On the other hand, Searles and von Nagy-Felsobuki [ 6 ] have developed the most general form of the EckartWatson Hamiltonian applicable non-linear molecules with small amplitudes of vibration and have cast the Watson operator in terms of a Taylor series expansion (as suggested by Watson [ 23 ] ) . The C, rovibrational Hamiltonian is based on rectilinear displacement coordinates and so collapses to Carney et al.% D3,, [ 14- 16 ] and CZv Hamiltonians [ 27 1. It has been demonstrated that provided the same force field was used, the scattering coordinate Hamiltonian and solution algorithm of Henderson et al. [22] yielded essentially the same vibrational band origins near the potential energy minimum as the normal coordinate approach developed by von NagyFelsobuki and co-workers [ 2 1,22 1. These vastly different solution algorithms yielded the first ten vibrational band origins of both ‘Liz and 6Li7LiT to within 0.03 cm-’ [ 21,221. Hence both algorithmic approaches have properly converged in describing small amplitudes of vibration. As an extension of our earlier work on the vibrationalpropertiesofLi: [6,7,18-21],Li2Na+ [6,7], LiNa$ [6,7], KLiNa+ [6,7], K*Li+ [8] and Nag [ 281, we report here ab initio variational calculation of the rotational energy levels of the low-lying vibrational states of the ground electronic state of Li2K+. Although the PE surfaces are speculative, the motivation underlying this investigation is to encourage and moreover, possibly assist experimentalists in detecting and identifying the rovibrational structure of Li*K+.
2. Potential energy surface of the ground electronic state of Li2K+ Ab initio electronic calculations were performed using the GAUSSIAN 88 suite of programmes [ 291 within the SDCI/FC ansatz. Size-consistency cannot
be properly accounted for using Davidson’s correction [ 301, due to the frozen-core approximation (which fixes the inner-shell orbitals, thereby reducing the calculations to a two-electron correlation problem ). For lithium, the ( 1 ls3pld/6s3pld) basis of Gerber and Schumacher [ 3 I] was used, with the partially optimized d exponent of 0.15 [6,7,18,19]. For potassium, the primitive basis of Huzinaga and Klobukowski [ 321 (20~13~) was used supplemented with d and f polarization functions (with partially optimized exponents of 0.16 and 0.09 respectively) giving a (20s13pldlf/l5s8pldlf) contracted basis set. This basis yielded an atomic energy of -599.1641718 E,, for the *S state, whereas a (20s13pld/15s8pld) and (20s13p2d/15s8p2d) yielded -599.1641543 and -599.1641548 E,, respectively. The energy improvement by adding an additional f function is significantly larger than increasing the number of d functions, although the number of d functions are far from saturated. Ideally, a much larger basis set is required for both lithium and potassium in order to ensure spectroscopic predictability for large rectilinear displacements from the minimum energy geometry. However, in order to optimize Li2K+ geometry using the Li: ( 11 s3pld/ 6s3pld) andK: (20s13pldlf/l5s8pldlf) basissets and employing the standard defaults in GAUSSIAN 88 took in excess of 5000 minutes of a VAX 3 100 workstation. Hence, in order to make the computations tractable and for small rectilinear displacements, similar basis sets have been used for generating the PE surfaces of Liz [ 6,7,18-211, Li,Na+ [6,7], LiNa: [6,7], KLiNa+ [6,7], K,Li+ [8] and Na: [ 281. In particular, for the latter, excellent agreement has been obtained for spectroscopic constants obtained by Carter and Meyer [ 331 using a pseudopotential-CI surface. A 73 point discrete SDCI/FC PE surface was constructed for the ground state (‘Ai) of Li2K+. Data points were selected near the potential energy minimum and most were made coincident with the quadrature points required by our potential energy integrator (which is the Harris-Engerholm-Gwinn or HEG quadrature scheme [ 34 ] ) , thereby minimizing the errors of the tit at these points. An initial set of discrete points was calculated along the diagonal of each vibrational coordinate. To these sparse number
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physics 172 (1993) 247-258 Table 1 Discrete potential Li2K+
249
Table 1 (Continued) energy surface of the ground electronic
state of
&L, (au)
RKL, (au)
&A,, (au)
E (Et,)
7.5301 7.6853 7.3749 7.9181 7.1421 8.3061 6.7540 8.6941 7.5155 7.5472 7.4984 7.5777 7.483 1 7.6412 7.4841 7.5014 7.5845 7.3910 7.8779 6.8350 8.5741 7.6707 7.3603 7.3920 8.2561 8.4143 6.7165 7.5462 7.5140 7.7982 7.3761 7.6860 6.7832 8.6719 7.1524 7.5369 7.6199 7.5358 6.7616 8.9192 7.6077 7.4252 7.8405 7.2197 8.0733 6.9868 7.5225 7.5383 7.5035 7.5669 7.4903
7.5301 7.6853 7.3749 7.9181 7.1421 8.3061 6.7540 8.6941 7.5155 7.5472 7.4984 7.5777 7.4831 7.6412 7.4841 7.5014 7.5845 7.6692 7.1824 8.2259 6.4879 7.6707 7.3603 7.3920 8.2561 8.4143 6.7165 7.8244 7.2358 5.7120 7.6549 7.4085 8.1829 6.6166 7.8505 8.2353 8.3112 6.8450 7.4602 6.8199 7.6077 7.4525 7.8405 7.2197 8.0733 6.9868 7.5225 7.5383 7.5035 7.5669 7.4903
5.3354 5.4454 5.2254 5.6103 5.0605 5.8852 4.7855 6.1602 5.5843 5.0865 5.9576 4.7131 6.5799 4.0909 7.2022 7.8244 9.0689 5.3412 5.3716 5.4786 5.6525 5.6943 5.4743 4.9765 7.1298 4.6407 6.6523 5.4510 5.2313 5.1367 5.5898 5.0926 6.6966 4.4967 7.8491 6.2636 5.0267 4.48 16 5.7167 7.1045 5.3904 5.2804 5.5553 5.1154 5.7203 4.9505 5.4598 5.2109 5.8332 4.8376 6.2065
-613.944625 -613.944491 -613.944445 -613.943788 -613.943482 -613.941566 -613.939555 -613.938396 -613.944281 -613.944175 -613.942708 -613.941447 -613.938473 -613.928519 -613.933521 -613.928752 -613.921285 -613.944543 -613.944105 -613.942242 -613.939456 -613.943882 -613.944399 -613.943595 -613.932729 -613.938678 -613.934390 -613.944414 -613.944358 -613.932510 -613.944186 -613.944113 -613.935731 -613.933187 -613.928156 -613.940164 -613.943005 -613.936928 -613.942372 -613.931338 -613.944595 -613.944577 -613.944083 -613.943904 -613.943038 -613.942298 -613.944539 -613.944514 -613.943345 -613.942680 -613.941182
7.6012 7.4605 7.7388 7.2519 7.0434 7.6777 7.7800 7.3540 7.7419 6.9424 8.3758 7.5381 7.4686 7.7548 7.8485 7.9342 8.7692 8.2068 7.5727 7.7852 7.5304 7.4852
of points
7.6012 7.5996 7.3215 7.8083 8.0171 7.6777 7.7800 7.3540 7.7419 6.9424 8.3758 7.6772 7.7468 7.6157 7.9876 8.2124 7.3782 6.8120 7.7097 7.3705 7.6697 7.9020
a preliminary
4.4642 5.3368 5.3484 5.3586 5.4060 5.5698 5.2514 5.5988 5.8737 6.1950 5.0141 5.3918 5.3961 5.4468 5.6117 5.7257 5.8541 5.8428 4.0928 4.7279 5.5162 5.3340
surface
-613.937787 -613.944605 -613.944440 -613.944294 -613.943593 -613.944262 -613.944390 -613.944111 -613.943035 -613.939603 -613.942099 -613.944575 -613.944515 -613.944572 -613.943771 -613.942973 -613.941354 -613.941328 -613.928555 -613.941410 -613.944413 -613.944387
was obtained
8000 HEG quadrature
and
used
points. Electronic energies were calculated at a vastly reduced subset. Additional points were selected within this grid to more precisely define the force field in the regions of “poor” fit. Hence, the final energy grid calculated yielded electronic energies close to the quadrature points chosen by the HEG scheme. Table 1 gives the full discrete potential energy surface of LizK+. For rovibrational calculations it is important to obtain force fields that accurately interpolate the surface between calculated points. Various power series expansions [ 35-381 and PadC approximant expansions [ 35,391 have been used in rovibrational calculations. In particular, the embedded force fields using expansion variables of Dunham [ 361, Simon, Parr and Finlan [ 37 1, Ogilvie [ 3 8 ] as well as their exponential variants [ 35,391 have been successfully employed. Of the analytical representations investigated, a fifth-order Ogilvie power series expansion gives the “best” tit to the discrete surface. Using SVD it is possible to ensure that the analytical PE surface is free from singularities in the integration region [ 35 1. Table 2 gives the analytical representation of the PE surface of L&K+, which was used in subseto generate
250 Table 2 Fifth-order
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physics 172 (I 993) 24 7-258
Ogilvie analytical Expansion
representation variable
I Pl +Pz P3 P: +p: P: PIP2 P2Ps+PIP3 P: +p: P: PTP2 +P:PI PTP, +P:P, PIP: +P*P: P1P2P3 P: +p: P: P:Pz +P:PI P?P, +P:Ps (x2)“*=2.07x
b,
of the potential
energy surface of Li2K+ a)
Coefficient
Expansion
-613.94463 0.00000 0.00000 0.11496 0.18112 -0.03176 -0.00633 -0.16996 -0.19246 0.00198 -0.08911 -0.00164 0.16268 -0.24741 -0.00937 0.03978 0.31989
PIP: +P*P: P:P: P:P: +P:P: P:PzP3+PIP:P3 P*PzP: Pi’+P: Pi: PTP2 +P:Pr PYPS +P:P3 PIP: +PxPf: P’;Pi +P:P: P’:Pi +P:P: P:P: +p:p: PilPzP, +PIP:P, PIPZP: PTPZPI P:P*P: +P,P:P:
variable
b,
Coefficient -0.01953 -0.14775 0.04459 0.13761 0.19235 -0.15369 0.06355 -0.10163 0.17492 -0.13920 -0.05523 0.12259 0.78913 -0.05953 -0.03397 0.13696 -0.21469
lo-“
‘) SVD analysis is used [ 351 with singular values q4 and q6_32 being set to zero. b, Ogilvie expansion variable has the form 2 (R, - R,) / (R, + R,) , where R, and R, are the instantaneous respectively.
quent vibrational and rovibrational calculations. Furthermore, crZ4 and 0Z26_32singular values were zeroed, since these expansion terms unrealistically affect the magnitude of the high-order coefficients without significantly affecting the (x2)1/2 value. Fig. 1 gives the contour plots of the PE surface in terms of rectilinear vibrational coordinates (labelled as t coordinates [ 27 ] ).
3. Rovibrational Hamiltonian and solution algorithm The vibrational Hamiltonian used in this investigation is the Eckart-Watson Hamiltonian developed in t coordinate space by Carney et al. [ 271. The Hamiltonian has the form
and equilibrium
bond distances
where F,. is the vibrational kinetic energy operator, p, is the vibrational angular momentum operator, ow the Watson operator (which is a mass dependent contribution to the potential energy operator) and P is the electronic potential energy operator. The Watson operator is the sum of the diagonal elements of the reciprocal effective moment of the inertial tensor. A perturbation expression for this operator (which is valid only for small amplitudes of vibration) has been derived with respect to t coordinates [ 8 1. It was incorporated in the vibrational Hamiltonian, thereby avoiding the embedded singularities [ 8 1. The ab initio variational solution of the CzVvibrational Hamiltonian can be described as follows. Onedimensional wavefunctions are calculated using a finite-element solution of a one-dimensional Hamiltonian, which is expressed in terms of a single rectilinear coordinate and so incorporates only the firstorder expansion of the Watson operator. The firstorder expansion is diagonal in the t coordinates. For each t coordinate, 1000 finite-elements were constructed within the following domains: t, [ - 3.0,4.2], t2 [ -2.2, 4.81, t3 [ -4.0, 4.01. A three-dimensional
F. Wang and E.I. von Nagy-Felsobukl /Chemical Physm 172 (1993) 247-258
251
4, 35.
(4
3=-
@I
3-
3-
25.
25.
2. 2lS. :
15. l-
::
l055-
;
0.5,
I
O-
0.
. 05. . -1. -lS-
-05. -1. I . II
-2. b -25.
-3-5. -2. -25. 3.
3-
Fig. 1. Two-dimensional potential energy contour plots for the 5th order power series expansion using an Ogilvie expansion variable. The singular values uZ4and u26_S2are zeroed.
configuration basis was constructed from the three one-dimensional solutions [ 13 1. The configuration list was pruned by using a nodal cut-off criterion [ 15 ] ; that is, all products containing more than 13 nodes were excluded and so a total of 560 basis functions were employed. For the three-dimensional vibrational Hamiltonian, the third-order expansion of the Watson operator was used. All integrals were evalu-
ated using a sixteen point Gauss quadrature scheme, except the potential energy integrals which were evaluated using the HEG scheme [ 6-8 1. Finally, a secular determinant was constructed using eq. ( 1) and spanned by the configuration basis. It was diagonalised to yield vibrational wavefunctions and eigenenergies, both of which are required for the rovibration problem. Table 3 assigns the 10 lowest-lying
F. Wangand E.I. vonNagy-Felsobuki/Chemical PhysicsI72 (1993) 247-258
252
Table 3 Assigned vibrational band origins of Li*K+ (cm-‘)
000 100,010 001 200,110 101,011 002 010,100 300,210 201,111,101 002, 102
Symmetry
% Weight a)
Vibrational band origin b,
Al Al B1 Al BI A, Al Al B, Al
95 67, 18 91 44,20 57,14 82 71,22 27, 16 32, 13, 13 41,13
0.0 120.2 129.8 237.4 245.8 257.1 305.7 325.5 359.1 369.5
“~%Weight={C~/CC~}‘~*XlOO. b, The zero-point energy of Li2K+ 1s280.6 cm-‘.
vibrational band origins of L&K+. The rovibrational Hamiltonian in vibration trix representation is given by [ 27 ]
ma-
A~R=E,(S)~+O.~(A)~~~~+O.S(B),,~~ +o.5(c),,r41+o.5(D>,(14x14y+nyz7x) + (ilfi) (F)&,
(2)
where E, is the ith pure vibrational eigenenergy, (S) I, is the overlap vibration matrix element and l?‘s are the rotational angular-momentum operators, whose components refer to the molecule-fixed coordinate system. In general, matrix elements involving the rotational and centrifugal distortion operators are given
matrix elements spanned by lowest five vibrational eigenfunctions. The 0.5 factor in eq. (2) is incorporated in the matrix elements labelled AI-D’. The matrix elements spanned by plus and minus combinations of the regular symmetric top eigenfunction (RJ’km) have been detailed by Carney et al. [ 27 1. The advantage of this basis is that the corresponding matrix elements spanned by the angular momentum operators are real. Diagonalisation of the supermatrix HVR yields the rovibrational wavefunctions and eigenenergies. Table 5 gives the calculated rotational eigenenergies up to J= 3 for the low-lying vibrational states. In order to ensure convergence of the calculated eigenenergies with respect to the vibrational basis set used, a number of truncated basis sets were investigated. Calculations were employed using up to ten vibrational eigenfunctions and up to the same J level. It was found that using five and ten vibrational wavefunctions the mean difference for all the rotational levels is 0.001 cm-‘. The definition of the reduced Hamiltonians has been given by Watson et al. [ 401. Vibrational constants (such as the fundamental frequencies and anharmonic constants) could not be obtained within reasonable precision using a simple least-squares fitting procedure, due to the “mixed” nature of the configuration wavefunctions of the higher-lying states. Nevertheless, following the usual prescription [ 4 1,42 1, the spectroscopic constants were obtained frorn least-squares fits of rotational energy levels of the low-lying live vibrational states. The constants are given in table 6.
by
(C>,=(ilbzlj),
(W,=(ilPxyIj)
,
where i, j are the indices of the vibrational eigenfunctions and ,u is the instantaneous effective reciprocal inertia tensor. A Taylor series expansion is used for p [ 81 to circumvent the numerical singularities. The A-D matrices are symmetric with the (0) and (A) (B) matrix elements coupling adjacent odd or even values of K. The F matrix represents the Coriolis coupling that splits levels with non-zero K values (i.e. z axis angular momentum component ) in the presence of vibrational angular momentum. Table 4 gives rotational, centrifugal distortion and Coriolis coupling
4. Dipole moment surface Difficulties were encountered using the GAUSSIAN 88 package in order to generate a CI dipole moment surface #‘. Hence, the discrete dipole moment surface of the ground electronic state of L&K+ were performed at the Hartree-Fock SCF level. A 68 point dipole moment surface was generated in terms of the #’ In the August 1992 Bulletin of the Ohio Supercomputer Service, Dr. M. Frisch suggested to use the KEYWORD FORCE in GAUSSIAN 88 in order to obtam a CI dipole moment surface. However, due to limitations on our disk storage capacity this was not feasible using such a large basis set.
F. Wang and E.I. van Nag-yFelsobuki / Chemxal Physics I72 (I 993) 247-258 Table 4 Rotatronal
and Coriolis matrix elements
253
of Lr2K+ (cm-r)
Matrix element
A’
B’
C’
D’
F .’
11
0.6018 -0.004s 0.6016 0.0000 0.0000 0.6080 -0.0019 - 0.0065 0.0000 0.6016 0.0000 0.0000 -0.0054 0.0000 0.6084
0.1169 0.0072 0.1160 0.0000 0.0000 0.1158 0.0009 0.0101
0.0976 0.0049 0.0968 0.0000 0.0000 0.0965 0.0005 0.0069 0.0000 0.0960 0.0000 0.0000 0.0049 0.0000 0.0957
-0.4965x lo-“’ 0.9382x 10-r” -0.9002x lo-” 0.1445x 10-l -0.2474x 1O-2 -0.2608x lo-” 0.1597x10-‘0 0.1597x10-9 0.2073x lo-* -0.1360x 1O-9 0.1683x lo-* 0.1440x 10-l -0.1642x 1O-9 -0.4007x 1o-2 0.4788X 10-10
0.2702x lo-l9 -0.1480x 10-s 0.4339x lo-‘8 -0.6649x lo-’ -0.8586X 10-l 0.1928x lo-‘s -0.3184x 10-10 -0.3368x 1O-9 0.7090x 10-z 0.7363x lo-‘s -0.3010x10-* -0.1463x 10-l -0.13%3x10-9 -0.1089 0.1290x 10-l’
21 22 31 32 33 41 42 43 44 51 52 53 54 55
0.1151 0.0000 0.0000 0.007 1 0.0000 0.1148
‘) The F matrix elements have the relationship Table 5 Rotational
F( I, j) = - F(J, I).
energy levels of L&K+ for low-lying vibrational
J K.
10 11 11 20 21 21 22 22 30 31 31 32 32 33 33
Kc
1 1 0 2 2 1 1 0 3 3 2 2 1 1 0
states (cm-‘)
E” 0.0000
120.1477
129.8307
237.3582
245.8418
0.2145 0.6994 0.7187 0.6428 1.1090 1.1670 2.6217 2.6223 1.2839 1.7229 1.8390 3.2652 3.2679 5.7384 5.7385
0.2136 0.6992 0.7 176 0.6400 1.1080 1.1631 2.6202 2.6209 1.2786 1.7205 1.8308 3.2612 3.2637 5.7361 5.7362
0.2115 0.7038 0.7239 0.6338 1.1068 1.1670 2.6439 2.6445 1.2654 1.7107 1.8307 3.2780 3.2817 5.7906 5.7906
0.2124 0.6991 0.7166 0.6366 1.1066 1.1591 2.6191 2.6199 1.2723 1.7166 1.8219 3.2570 3.2593 5.7352 5.7358
0.2090 0.7027 0.7232 0.6261 1.1001 1.1616 2.6428 2.6436 1.249 1 1.6956 1.8178 3.2689 3.2735 5.7902 5.7903
rectilinear displacement coordinates #*. The grid used for the dipole moment surface differed from that used for the electronic energy calculations only with respect to its size (68 points instead of 73). Points located well away from the minimum energy geometry (i.e. rectilinear displacements larger than 2.0 au) were O2 Discrete authors.
dipole
moment
surface
available
on request
from
excluded in order to facilitate a more precise fit. The geometries were rotated and/or reflected at each data point in order to coincide with the Eckart frame, thereby that the dipole moment are computed with respect to the centre of mass. The molecule was placed in the xy plane, with the origin coinciding with the centre of mass and the positive x axis bisecting the included angle for the CzV symmetry.
F. Wang and E.I. von Nagy-Felsobukr /Chemical Physrcs 172 (1993) 247-258
254 Table 6 Spectroscopic
constants
of L&K+ for the lowest five vibrational
lko
v=l
Icar
-0.9234
A+C A-C rrsssan
0.2097x 0.1512x -0.0866 0.0792 -0.1823 -4.1984 -0.1031 4.5672
7cccc
rAAcc rX%cc 7AABB
reduction A, AJK AK 6, & first-order DJ DJK DK 6, & R6
states (MHz) v=2
-0.9240 10’ 105
0.2095x 10’ 0.1512~10~ 2.8446 -2.4594 2.8663 78.4762 1.2003 - 78.4480
constants 0.0009 -0.0692 0.1138 0.0101 -0.1251
v=3
-0.9245 0.2075 x 10’ 0.1578X105 -22.3850 -22.3176 - 17.7130 -138.2331 - 20.4923 558.0445
v=4
-0.9244 0.2090x 10’ 0.1510x10’ 15.2205 -4.4081 5.4620 149.3292 2.3855 - 146.2992
-0.9255 0.2006x lo5 o.1660x105 - 62.3548 - 70.9043 - 52.9245 -378.3192 -62.6884 1574.1887
distortion
centnfugal distortion -0.2848 1.6449 - 1.3146 0.0104 -0.2458 -0.1428
a) Ray’s asymmetry
-0.0481 -0.1627 -0.5057 -0.3315 -2.1921
5.5878 - 116.5932 115.433’7 0.0042 135.0186
-1.3515 2.7008 -2.7147 - 1.2268 -6.5668
16.6574 -333.2675 329.8412 -0.5343 380.4753
4.8669 -29.6529 24.0694 -0.3315 6.3996 2.4575
- 30.6869 101.0552 -65.9400 0.0042 - 106.6288 -18.1374
a. 1300 -54.1888 44.6932 - 1.2268 13.5095 4.7408
-85.8938 282.0394 - 182.9145 -0.5343 300.7253 -51.2756
constants
parameter.
For a charged triatomic molecule of CZVsymmetry, the p,, component at the equilibrium geometry is zero, whereas there is a permanent dipole moment for the pX component. A dipole moment surface can be generated using a power series expansion of the rectilinear coordinates [ 81. Table 7 gives the calculated regression coefficients of Li2K+ using the t coordinate expansion. From the coefftcients it is clear that the surface is dominated by the linear term (as the C, coefftcient is the largest). Nevertheless, the nonlinear terms are of significant magnitude suggesting that electrical anharmonicity cannot be neglected. The vibrational eigenenergies, wavefunctions and dipole moment surface were used in order to calculate the dipole moment matrix elements, Einstein transition probabilities (A,,,,, and II,,), band strengths (S,,,) and vibrational radiative lifetimes (7). Table 8 details these quantities together with the calculated transition frequencies for the low-lying vibrational states. Using the variational rovibrational wavefunctions, the dipole transition matrix elements are calculated
Table 7 Expansion
coeffkrents
for dipole momentum
surface of Li2K+ ‘)
Expansion variable
PX
Expansion variable
PY
G t1+ r2 (tt+f2Y t: (t,+t*)3 t:(t, +t21 (t, +tzY t: t:(tt +t*Y
0.83152 0.25788 0.08767 -0.02544 -0.03571 0.43869 -0.02987 0.07968 0.09000
ts f3(rt+f2) t: ts(f,+f2Y &t, +t21 t:(t, +r2) t3(ct+W t: f3(f,+f*Y
0.50276 0.47137 -0.00628 0.13353 -0.01253 -0.16115 0.00917 0.06001 -0.10591
0.01238 (ft+f2Y t%t, +tz1 -0.14193 t:(t, +t*Y -0.03616 p) All entries in atomic units.
&t* +f*Y
and utilised to calculate the transition matrix elements involving individual line intensities [ 43,441. Table 9 gives the variationally calculated rovibra-
F. Wang and E.I. von Nagy-Felsobukz / Chemrcal Physics I72 (I 993) 247-258
255
Table 8 Vibrational transition frequencies, square dipole moment matrix elements, Einstein coefficients ‘), band strengths ‘) and radiative lifetimes for Li2K+
i
1
1 0 20 30 40 5 0 6 0 70 80 9 0
V/l
fi;
4
4
41
(cm-‘)
(D2)
(s-l)
( 1016cm3erg-’ s-‘)
(atm-’ cm-*)
Tb) (s)
120.15 129.83 237.36 245.84 257.08 305.67 352.45 359.07 369.54
0.72x 1O-2 0.12 0.45x 1o-4 0.30x 1OP o.19x1o-3 0.26x 10-l 0.21 x 1o-4 0.70x 1o-5 0.57x 1o-4
0.0039 0.0799 0.0002 0.0001 0.0010 0.2311 0.0003
1.3584 21.9348 0.0085 0.0056 0.0352 4.8584 0.0039 0.0013 0.0107
9.6962 169.1836 0.1192 0.0815 0.5376 88.2262 0.0816 0.0280 0.2344
2.14 1.63 2.80 1.61 2.45 2.87 8.05 3.50 2.46
a) Calculated at 300 K. b, Lifetime of the upper state. Table 9 Variationally calculated rovibrational absorption line intensities (at 300 K) for X state of L&K+ VI J’ K:, K:
u” J”
K”P K”e
Branch P,
0 44 0 44 0 54 1 11 132 120 2 20 2 32 2 30 3 11 3 22 3 30 4 22 431 4 33
0 0 1 1 2 2 2 1 3 1 0 3 1 3 0
0 0 042 0 032 0 0 0 0 022 0 022 032 0 0
5 4
0 1
2
1
1 3 3 2
0 1 2 1
2
1
3 2
1 1
5 3 3 2 1 1 3 2 2 1 2 1 1 2 2
Q, R
-1 0 1 -1 0 -1 0 1 -1 0 1 -1 0
V4xB) (cm-‘)
.s,b) (atm-’ cm-‘)
Rk
‘)
6.86 7.32 7.01 119.74 120.14 120.57 128.74 129.85 129.99 235.44 238.87 236.01 245.22 245.70 250.52
0.210x 10-r 0.250~ 1O-2 0.265x lo-* 0.950x 10-3 0.933x 10-s 0.226x lo-’ 0.532x lo-* 0.141x10-’ 0.151x10-’ 0.180x 1O-4 0.184x 1O-4 0.180x 1O-4 0.125~ 1O-4 o.121x1o-4 0.131x10-4
0.378x 10’ 0.393x 10’ 0.458~ 10’ 0.717x lo-2 0.707x 10-Z 0.167~10-~ 0.355x 10-l 0.937x 10-l 0.988x 10-l 0.449x 1o-4 0.447x 1o-4 0.448x 1O-4 0.293x 1O-4 0.282x 1O-4 0.295x 1O-4
(Da)
‘) Difference of eigenenergies between rovibrational states labelled as A and X, where A and X represent the upper and lower rovibrational states. b, Band strengths between rovibrational states labelled as A and X, where A and X represent the upper and lower rovibrational states. ‘) Square of the dipole matrix element spanned by the rovibrational states are labelled as AX, where A and X represent the upper and lower rovibrational states.
tional
absorption
lected number
line intensities
of Li2K+ for a se-
of transitions.
5. Discussion A number of pseudopotential MO calculations have been performed on the electronic ground state of
L&K+ using small basis sets. Pseudopotential models do not take into account core-valence or core-core effects in a rigorous manner. On the other hand, we have used sizeable basis sets within the all-electron SDCI/FC ansatz. At small displacements from the equilibrium geometry, the single reference treatment is justified if the leading coefficients are large and nearly constant. This was shown to be the case for
256
F. Wang and E.I. van Nagy-Felsobuki /Chemical Physrcs 172 (1993) 247-258
alkaline-earth oxides, fluorides and hydroxides [ 45 1. On the other hand, the FC approximation is less satisfactory on theoretical grounds, even though it is tractable for more electron-dense systems. Nevertheless, comparison of the low-lying vibrational band origins of Li: using SDCI/Full with the SDCI/FC [ 6,7,19] gives some credence to this approximation at small rectilinear displacements. The SDCI/FC surface predicts Li2K+ minimum energy structure to be of CzV symmetry, with energy of -613.944625 E,,. The predicted equilibrium structural parameters ( RLIK, LiKLi bond angle) are (7.53 ao, 41 .SO), which compares well with pseudopotential-C1 value of (7.02 ao, 45.7” ) [ 5 1. Both calculations however differ more significantly with the Hartree-Fock pseudopotential value of (6.30 ao, 4 1.8’ ) [ 41. Similar differences between SDCI/FC [ 6-8 ] and pseudopotential-CI calculations [ 5 ] are shown in the structural parameters for the other Czy isovalent alkali metal cations. For example, for Li,Na+, LiNa: and K,Li+ absolute differences are (0.20 ao, 3.5”), (0.16 ao, 5.3”) and (0.23 ao, 6.7”) respectively [ 5-81. It should not be forgotten that core-core and core-valence interactions cannot be rigourously modelled in pseudopotential formalism [48-501 and so the total energy and therefore the predicted geometries are sensitive on the definition of the valence and core sub-spaces. The power series with the smallest x2 is not necessarily the “best” fit to a PE surface. It is important that the analytical representation is consistent with anticipated physical properties [ 12,13,18,35 ] and moreover, is smooth in the integration region, since numerical integration schemes are used to evaluate potential energy integrals. High-order power series are often problematic, because of singularities in the domain under consideration [ 12,35 1. In order to detect singularities, numerous graphical inspections need to be performed. Fig. 1 demonstrates that the analytical surface is smooth everywhere, with monotonically increasing repulsive walls within the integration region. The force field given in table 2 was used in the subsequent rovibrational calculations, since it provides the most accurate and smooth interpolation function. The fifth-order Ogilvie power series yields a (x2)l12 of 2.07x lo-‘, whereas for K,Li+ [S] a sixthorder exponential Dunham expansion variable was
employed yielding a (x2) ‘I2 of 2.57~ 10T4. On the other hand, for Li,Na+ and LiNa> [ 6 1, PadC force fields of order P (6,4 ) and P (4, 5 ) yielded the smallest (x2) ‘I*. Hence, the use of different expansion variables and power series of the force fields of these CzVisovalent molecules, suggests that there are subtle differences in their PE surfaces even close to the minimum. Table 3 gives the calculated vibrational band origins up to 370 cm-‘. As full mechanical anharmonicity is embedded in the vibrational Hamiltonian as well as operators coupling the t vibrational modes, the vibrational assignment is no longer simple, since mixing can occur for configurational basis functions belonging to the same irreducible representation. For example, configurational basis functions such as (000), (loo), (110) and (002) are all of Ai symmetry and so can have non-zero coefficients in the linear expansions of vibrational eigenfunction. Table 3 highlights this point by giving the percentage weight [ 6- 8,12,13,17-2 1 ] of the dominant configurational basis functions. It is clear that the harmonic approximation (within the framework of uncoupled modes), whilst intuitive, is too simplistic for interpretation of the dynamics of this molecule. The sequence of the five lowest-lying vibrational band origins of Li2K+ (given in table 3) is: (000)<(100)<(001)<(200)<(101).Thisisbasically consistent with the assignment for LiNa$ [ 6,7], although the fifth vibrational band origin of the latter is assigned as (0 10) and the ( 10 1) is assigned as the sixth. For Li,Na+ [6,7] the same sequence has the (0 10) vibrational band origin of lower energy to the (200) vibrational band origin, whereas for K,Li+ [ 8 ] the vibrational eigenfunctions are far more mixed, since configurational basis belonging to the same irreducible representation are nearer in energy and so can more strongly interact. It is anticipated that a larger configurational basis set expansion should not alter the sequence of the live lowestlying vibrational band origins, since for Li2K+, Li,Na+ [ 6,7] and LiNa: [ 6,7] the percentage weights of configurational basis functions in the vibrational eigenfunctions are large for most of these vibrational states. As expected, far greater mixing of the basis functions occurs for the vibrational eigenfunctions at higher energies and/or for the heavier isovalent molecules (such as for K2Li+ [ 8 ] ). Hence,
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physics 172 (I 993) 247-258
larger configurational basis expansions may alter the sequence of these higher-lying vibrational band origins for all of the isovalent alkali metal cations [ 6-
81. The rotational, centrifugal distortion and Coriolis matrix elements spanned by the lowest five vibrational states are given in table 4. The rotational constants for the ground vibrational state are 0.602,O. 117 and 0.098 cm-’ respectively. The centrifugal distortion constants are expected to be small near the potential energy minimum and the calculations reflect this anticipation by yielding values of the order of lo-” cm- I. The largest centrifugal distortion constants are those spanned by wavefunctions belonging to different irreducible representations. These elements are of magnitude of 10-l to 10m2 cm-‘. Similar observations can be made for the Coriolis matrix elements. However, it should be noted that the Coriolis operator for an Eckart-Watson Hamiltonian yields numerically small diagonal matrix elements when compared with wavefunctions obtained from less diagonal Hamiltonian (s). The diagonal Coriolis elements are typically of order 1O-l9 to 1O- ” cm-‘, whereas the largest element is cs4, which has value of -0.1 cm-‘. For Li&+, the limiting case for the rotational levels is Mulliken’s prolate symmetric top, which is reflected by the calculated value of Ray’s asymmetry parameter of -0.92 for the lowest-lying five vibrational states. Hence, in table 5 the rotational energy levels up to the J= 3 level are assigned for the lowestlying five vibrational states, within the framework of a prolate symmetric top. Table 6 details the calculated rotational spectroscopic constants. obtained from our ab initio force fields and moreover, using our ab initio rovibrational states fitted to reduced Hamiltonians [ 40-421. High resolution rovibrational spectra are usually assigned in this manner, but using experimentally measured rovibrational levels. Although we have calculated the quartic centrifugal distortion constants at the equilibrium structure and neglected spin-rotation interactions, the signs and magnitude of these constants should be of assistance in the spectroscopic detection of Li*K+. In order to calculate transition probabilities, band strengths and vibrational radiative lifetimes a discrete dipole moment surface is required. Generally,
251
there are too few discrete dipole moment surfaces in the literature in order to estimate reliably the magnitude of error associated with a dipole hypersurface. Nevertheless, Green [ 491 has concluded that for a neutral diatomic molecule with a single sigma bond, using a double zeta basis set augmented with polarisation functions the error associated with a dipole moment at the Hartree-Fock limit is of the order 0.1 to 0.2 D. The basis set we have employed for Li2K+ certainly meets this criterion, although it is deficient with respect to producing a reliable Hartree-Fock limit. It should be noted that Li2K+ is bonded via single sigma bonds. However, what is not known is the error variation of the dipole moment hypersurface over small amplitudes of displacements. Table 7 indicates that the linear expansion variable is dominant, although the anharmonic terms are too sizable to neglect. Hence, care must be taken, since the error may not be uniform over the entire hypersurface nor of the order of 0.2 D. Table 8 gives the Einstein transition probabilities, band strengths and vibrational radiative lifetimes with respect to transitions from the ground vibrational state to the ninth excited vibrational state calculated using vibrational eigenfunctions and eigenenergies. For the CIV point groups there are no Raman forbidden transitions, unlike Liz which has D3h symmetry. Hence the lifetimes of the excited vibratibnal states of K2Li+ are small for these transitions compared with those of Li: [ 6,7,19,21]. In order to determine rotational line intensities the rotational partition function was obtained from the variationally calculated rotational levels. Neglecting contributions from the vibrational partition function, the intensities were calculated at 300 K. It would be anticipated that some rovibrational transitions should be accessible using laser spectroscopy, even though no such experimental data are currently available. Table 9 gives the absolute line intensities and the squares of the electric dipole transition matrix elements between the vibrational ground state and lowest-lying four excited vibrational states for some of the most intense transitions within the P, Q and R branches. It is hoped that these calculations will assist in the rovibrational spectroscopic detection of this molecule and moreover, will promote even more extensive configuration interaction calculations.
258
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physrcs I72 (1993) 247-258
Acknowledgement All calculations were performed using the VAX 3100 and VAX 665Os, the latter made available by the generous support of the Computing Centre, The University of Newcastle. We also wish to acknowledge the support of the Research Management Committee, The University of Newcastle. Finally, we wish to acknowledge the Overseas Postgraduate Research Award held by Ms. F. Wang.
References [ 1] B.K. Rao and P. Jena, Intern. J. Quantum Chem. Symp. 22 (1988) 287. [2] J. Koutechy and P. Fantucci, Chem. Rev. 86 (1986) 539. [ 3 ] F. Wang, N. Andriopoulos, N. Wright and E.I. von NagyFelsobuki, J. Cluster Sci. 2 ( 1991) 203. [4] R. Gaspar and I. Tamassy-Lentai, Acta Phys. Acad. Sci. Hung.44 (1978) 191. [S] D. Pavolini and F. Spiegelmann, J. Chem. Phys. 87 ( 1987) 2854. [6] D.J. Searles and E.I. von Nagy-Felsobuki, J. Chem. Phys. 95 (1991) 1107. [ 71 F. Wang, D.J. Searles and E.I. von Nagy-Felsobuki, J. Mol. Struct. THEOCHEM 272 ( 1992) 73. [ 81 F. Wang, D.J. Searles and E.I. von Nagy-Felsobuki, J. Phys. Chem. 96 (1992) 6158. 191 M.M. Kappes, M. Schlar, P. Radi and E. Schumacher, J. Chem. Phys. 84 (1985) 1863. [lo] C. Brechignac, Ph. Cahuzac and J.Ph. Roux, Chem. Phys. Letters 127 (1986) 445. [ 111 W.C. Stwalley and M.E. Koch, Opt. Eng. 19 ( 1980) 7 1. [ 121 D.J. Searles and E.I. von Nagy-Felsobuki, in: Vibrational spectra and structure, Vol. 19, ed. D.R. Durig (Elsevier, Amsterdam, 199 1) [ 13 ] P.G. Burton, E.I. von Nagy-Felsobuki, G. Doherty and M. Hamilton, Chem. Phys. 83 (1984) 83. [ 141 G.D. Camey and R.N. Porter, J. Chem. Phys. 60 (1974) 4251. [lS]G.D. Camey and R.N. Porter, J. Chem. Phys. 65 (1976) 3547. [ 161 G.D. Camey and R.N. Porter, Phys. Rev. Letters 45 (1980) 537. [ 171 G.C. Lie andD. Frye, J. Chem. Phys. 96 (1992) 6784. [ 181 S.J. Dunne, D.J. Searles and E.I. von Nagy-Felsobuki, Spectrochim. Acta A 43 ( 1987) 699. [ 191 S.J. Dunne, D.J. Searles and E.I. von Nagy-Felsobuki, Spectrochim. Acta A 44 ( 1988) SOS. [20] S.J. Dunne, D.J. Searles and E.I. von Nagy-Felsobuki, Spectrochim. Acta A 44 (1988) 985.
[21 I D.J. Searles and E.I. von Nagy-Felsobuki, Australian J. Chem. 42 (1989) 737. [22] J.R. Henderson, S. Miller and J. Tennyson, Spectrochim. Acta A 44 (1988) 1287. [23] J.K. Watson, Mol. Phys. 15 (1968) 479. [24] J.K. Watson, Mol. Phys. 19 (1970) 465. [25] J. Tennyson and B.T. Sutcliffe, Mol. Phys. 51 (1984) 887. [ 261B.T. Sutcliffe, in: Methods of computational chemistry, Vol. 5, ed. S. Wilson (Plenum Press, New York, 1990). [ 27 ] G.D. Camey, S.R. Langhoff and L.A. Curtiss, J. Chem. Phys. 66 (1977) 3724. [ 281 F. Wang and E.I. von Nagy-Felsobuki, Mol. Phys. 77 (1992) 1197. [29] M.J. Frisch, M. Head-Gordon, H.B. Schlegel, K. Ragavachari, J.S. Binkley, C. Gonzalez, D.J. DeFrees, D.J. Fox, R.A. Whiteside, R. Seeger, CF. Melius, J. Baker, R. Martin, L.R. Kahn, J.J.P. Stewart, E.M. Fluder, S. Topiol and J.A. Pople, GAUSSIAN 88 (Gaussian Inc., Pittsburgh, PA, 1988). [30] S.R. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. [ 3 11W.H. Gerber and E. Schumacher, J. Chem. Phys. 76 ( 1982) 3075. [32] S. Huzinaga and M. Klobukowski, J. Mol. Struct. THEOCHEM 167 (1988) 1. [ 331 S. Carter and W. Meyer, J. Chem. Phys. 93 (1990) 8902. [ 341 D.O. Harris, G.G. Engerholm and W.D. Gwinn, J. Chem. Phys. 43 (1965) 1515. [35] D.J. Searles and E.I. von Nagy-Felsobuki, Computer Phys. Commun. 67 (1992) 527. [36] J.L. Dunham, Phys. Rev. 41 (1932) 721. [37] G. Simons, R.G. Parr and J.M. Finlan, J. Chem. Phys. 59 (1968) 5199. [ 381 J.F. Ogilvie, Proc. Roy. Sot. A 378 (1981) 287. [ 391 D.J. Searles, Ph.D. Thesis, The University of Newcastle, Newcastle ( 199 1) [40] J.K.G. Watson, J. Chem. Phys. 48 (1968) 181. [41] M. Wolfsberg, A.A. Massa and J.W. Pyper, J. Chem. Phys. 53 (1970) 3138. [ 42 1H.W. Kroto, Molecular rotation spectra (Wiley, New York, 1975). [43] J.W.C. Johns, J. Mol. Spectry. 125 (1987) 442. [ 44] B. Weis, S. Carter, P. Rosmus, H.-J. Werner and P.J. Knowles, J. Chem. Phys. 91 (1989) 2818. [4S]H. Partridge, S.R. Langhoff and C.W. Bauschlicher Jr., J. Chem. Phys. 84 (1986) 4489. [46 1E.I. von Nagy-Felsobuki and J.B. Peel, J. Chem. Sot. Faraday. Trans. II 74 (1978) 2204. [47 1E.I. von Nagy-Felsobuki and J.B. Peel, Australian J. Chem. 31 (1978) 2571. [48 ] N. Andriopoulos and E.I. von Nagy-Felsobuki, Australian J. Phys. 41 (1988) 563. [ 49 ] S. Green, Advan. Chem. Phys. 25 ( 1974) 179.
Variational calculations of rovibrational states of Li,K+ F. Wang and E.I. von Nagy-Felsobuki
’
Department of Chemistry, The University of Newcastle, Callaghan. NSW 2308, Australia Received 19 October 1992
The low-lying rovibrational states for the ground electronic state of Li*K+ were calculated using an ab initio variational solution of the nuclear Schriidinger equation. A discrete configuration interaction potential energy surface was generated and an analytical representation was obtained using a power series expansion. This force field was embedded in the Eckart-Watson Hamiltonian from which rovibrational wavefunctions and eigenenergies were variationally calculated. An SCF dipole moment surface was generated and used to calculate absolute line intensities and square dipole moment matrix elements for some of the most intense transitions within the P, Q and R branches, between the vibrational ground state and the low-lying rovibrational states.
1. Introduction Theoretical studies on alkali cluster ions have centered on their electronic structure, stability and equilibrium geometry [ l-31. While most work has focussed on the homogeneous cluster ions [ l-3 1, more recent studies have been extended to the electronic structure of mixed clusters of form X,,Y + (X, Y = Li, Na, K) based on pseudopotential [ 4,5 ] and all-electron [ 1,6-81 molecular orbital methodologies. This has been partly prompted by the observations of the heterogeneous alkali cluster ions in supersanic expansions [ 9,101. So far no observatio?phave been made on their rovibrational s$_c&r& even though such information would be valuable with respect to their proposed significance in a number of technological processes [ 111. Recently, ab initio solutions of the “complete” molecular Schriidinger equation have considerably unravelled the rovibrational structure of small molecules [ 6-8,12-22 1. By “complete” we mean that an ab initio Born-Oppenheimer potential energy (PE) surface was generated and embedded in the nuclear Schriidinger equation in order to variationally yield rovibrational eigenvalues and eigenfunctions [ 12 1. In the most celebrated exthis regard, H: is perhaps ample, since quanta1 investigations correctly pre-
’ To whom correspondence should be addressed. 0301-0104/93/$06.00
dieted its infrared spectrum near the potential energy minimum prior to experimental observations [ 1217]. Due to lack of computing capacity, more speculative configuration interaction (CI) surfaces are usually entertained for electron-dense molecules. Simple CI truncations, such as single and double substitutions from a single Hartree-Fock reference determinant (SDCI) and basis set limitations are imposed in order to make computations tractable [6-8,18221. For molecules containing atoms beyond the first row, additional restrictions are imposed. For example, in the case of L&Na+, LiNa$, KLiNa+ and K,Li+ von Nagy-Felsobuki and co-workers [ 6-8,182 1] have restricted the CI subspace to only valence molecular orbitals (coined the “frozen-core” approximation and labelled FC ) . Alternatively, Pavolini and Spiegelmann [ 5 ] in their study of X2Y+ (X, Y = Li, Na, K) clusters, have resorted to a pseudopotential CI methodology, because of the need for a similar trade-off. Whatever the limitations, the motivation for these studies is to produce PE surfaces which are not only cost-effective, but spectroscopically predictive. Even for triatomic molecules, Eckart’s notion of an embedded equilibrium geometry may become problematic, since there are different rovibrational Hamiltonians for bent and linear nuclear configurations (as derived by Watson [23,24] ). A smooth transi-
0 1993 Elsevier Science Publishers B.V. All rights reserved.
248
F. Wang and E.I. von Nagy-Felsobuki /Chemrcal Physics 172 (1993) 247-258
tion cannot be made between the two Hamiltonians, because singularities occur in the mass-dependent potential energy operator (termed the Watson operator) of the bent Hamiltonian for linear configurations. Consequently, Tennyson and Sutcliffe [ 25,261 have developed a rovibrational Hamiltonian in terms of a body-fixed scattering coordinate system, in which there is no embedded equilibrium geometry. On the other hand, Searles and von Nagy-Felsobuki [ 6 ] have developed the most general form of the EckartWatson Hamiltonian applicable non-linear molecules with small amplitudes of vibration and have cast the Watson operator in terms of a Taylor series expansion (as suggested by Watson [ 23 ] ) . The C, rovibrational Hamiltonian is based on rectilinear displacement coordinates and so collapses to Carney et al.% D3,, [ 14- 16 ] and CZv Hamiltonians [ 27 1. It has been demonstrated that provided the same force field was used, the scattering coordinate Hamiltonian and solution algorithm of Henderson et al. [22] yielded essentially the same vibrational band origins near the potential energy minimum as the normal coordinate approach developed by von NagyFelsobuki and co-workers [ 2 1,22 1. These vastly different solution algorithms yielded the first ten vibrational band origins of both ‘Liz and 6Li7LiT to within 0.03 cm-’ [ 21,221. Hence both algorithmic approaches have properly converged in describing small amplitudes of vibration. As an extension of our earlier work on the vibrationalpropertiesofLi: [6,7,18-21],Li2Na+ [6,7], LiNa$ [6,7], KLiNa+ [6,7], K*Li+ [8] and Nag [ 281, we report here ab initio variational calculation of the rotational energy levels of the low-lying vibrational states of the ground electronic state of Li2K+. Although the PE surfaces are speculative, the motivation underlying this investigation is to encourage and moreover, possibly assist experimentalists in detecting and identifying the rovibrational structure of Li*K+.
2. Potential energy surface of the ground electronic state of Li2K+ Ab initio electronic calculations were performed using the GAUSSIAN 88 suite of programmes [ 291 within the SDCI/FC ansatz. Size-consistency cannot
be properly accounted for using Davidson’s correction [ 301, due to the frozen-core approximation (which fixes the inner-shell orbitals, thereby reducing the calculations to a two-electron correlation problem ). For lithium, the ( 1 ls3pld/6s3pld) basis of Gerber and Schumacher [ 3 I] was used, with the partially optimized d exponent of 0.15 [6,7,18,19]. For potassium, the primitive basis of Huzinaga and Klobukowski [ 321 (20~13~) was used supplemented with d and f polarization functions (with partially optimized exponents of 0.16 and 0.09 respectively) giving a (20s13pldlf/l5s8pldlf) contracted basis set. This basis yielded an atomic energy of -599.1641718 E,, for the *S state, whereas a (20s13pld/15s8pld) and (20s13p2d/15s8p2d) yielded -599.1641543 and -599.1641548 E,, respectively. The energy improvement by adding an additional f function is significantly larger than increasing the number of d functions, although the number of d functions are far from saturated. Ideally, a much larger basis set is required for both lithium and potassium in order to ensure spectroscopic predictability for large rectilinear displacements from the minimum energy geometry. However, in order to optimize Li2K+ geometry using the Li: ( 11 s3pld/ 6s3pld) andK: (20s13pldlf/l5s8pldlf) basissets and employing the standard defaults in GAUSSIAN 88 took in excess of 5000 minutes of a VAX 3 100 workstation. Hence, in order to make the computations tractable and for small rectilinear displacements, similar basis sets have been used for generating the PE surfaces of Liz [ 6,7,18-211, Li,Na+ [6,7], LiNa: [6,7], KLiNa+ [6,7], K,Li+ [8] and Na: [ 281. In particular, for the latter, excellent agreement has been obtained for spectroscopic constants obtained by Carter and Meyer [ 331 using a pseudopotential-CI surface. A 73 point discrete SDCI/FC PE surface was constructed for the ground state (‘Ai) of Li2K+. Data points were selected near the potential energy minimum and most were made coincident with the quadrature points required by our potential energy integrator (which is the Harris-Engerholm-Gwinn or HEG quadrature scheme [ 34 ] ) , thereby minimizing the errors of the tit at these points. An initial set of discrete points was calculated along the diagonal of each vibrational coordinate. To these sparse number
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physics 172 (1993) 247-258 Table 1 Discrete potential Li2K+
249
Table 1 (Continued) energy surface of the ground electronic
state of
&L, (au)
RKL, (au)
&A,, (au)
E (Et,)
7.5301 7.6853 7.3749 7.9181 7.1421 8.3061 6.7540 8.6941 7.5155 7.5472 7.4984 7.5777 7.483 1 7.6412 7.4841 7.5014 7.5845 7.3910 7.8779 6.8350 8.5741 7.6707 7.3603 7.3920 8.2561 8.4143 6.7165 7.5462 7.5140 7.7982 7.3761 7.6860 6.7832 8.6719 7.1524 7.5369 7.6199 7.5358 6.7616 8.9192 7.6077 7.4252 7.8405 7.2197 8.0733 6.9868 7.5225 7.5383 7.5035 7.5669 7.4903
7.5301 7.6853 7.3749 7.9181 7.1421 8.3061 6.7540 8.6941 7.5155 7.5472 7.4984 7.5777 7.4831 7.6412 7.4841 7.5014 7.5845 7.6692 7.1824 8.2259 6.4879 7.6707 7.3603 7.3920 8.2561 8.4143 6.7165 7.8244 7.2358 5.7120 7.6549 7.4085 8.1829 6.6166 7.8505 8.2353 8.3112 6.8450 7.4602 6.8199 7.6077 7.4525 7.8405 7.2197 8.0733 6.9868 7.5225 7.5383 7.5035 7.5669 7.4903
5.3354 5.4454 5.2254 5.6103 5.0605 5.8852 4.7855 6.1602 5.5843 5.0865 5.9576 4.7131 6.5799 4.0909 7.2022 7.8244 9.0689 5.3412 5.3716 5.4786 5.6525 5.6943 5.4743 4.9765 7.1298 4.6407 6.6523 5.4510 5.2313 5.1367 5.5898 5.0926 6.6966 4.4967 7.8491 6.2636 5.0267 4.48 16 5.7167 7.1045 5.3904 5.2804 5.5553 5.1154 5.7203 4.9505 5.4598 5.2109 5.8332 4.8376 6.2065
-613.944625 -613.944491 -613.944445 -613.943788 -613.943482 -613.941566 -613.939555 -613.938396 -613.944281 -613.944175 -613.942708 -613.941447 -613.938473 -613.928519 -613.933521 -613.928752 -613.921285 -613.944543 -613.944105 -613.942242 -613.939456 -613.943882 -613.944399 -613.943595 -613.932729 -613.938678 -613.934390 -613.944414 -613.944358 -613.932510 -613.944186 -613.944113 -613.935731 -613.933187 -613.928156 -613.940164 -613.943005 -613.936928 -613.942372 -613.931338 -613.944595 -613.944577 -613.944083 -613.943904 -613.943038 -613.942298 -613.944539 -613.944514 -613.943345 -613.942680 -613.941182
7.6012 7.4605 7.7388 7.2519 7.0434 7.6777 7.7800 7.3540 7.7419 6.9424 8.3758 7.5381 7.4686 7.7548 7.8485 7.9342 8.7692 8.2068 7.5727 7.7852 7.5304 7.4852
of points
7.6012 7.5996 7.3215 7.8083 8.0171 7.6777 7.7800 7.3540 7.7419 6.9424 8.3758 7.6772 7.7468 7.6157 7.9876 8.2124 7.3782 6.8120 7.7097 7.3705 7.6697 7.9020
a preliminary
4.4642 5.3368 5.3484 5.3586 5.4060 5.5698 5.2514 5.5988 5.8737 6.1950 5.0141 5.3918 5.3961 5.4468 5.6117 5.7257 5.8541 5.8428 4.0928 4.7279 5.5162 5.3340
surface
-613.937787 -613.944605 -613.944440 -613.944294 -613.943593 -613.944262 -613.944390 -613.944111 -613.943035 -613.939603 -613.942099 -613.944575 -613.944515 -613.944572 -613.943771 -613.942973 -613.941354 -613.941328 -613.928555 -613.941410 -613.944413 -613.944387
was obtained
8000 HEG quadrature
and
used
points. Electronic energies were calculated at a vastly reduced subset. Additional points were selected within this grid to more precisely define the force field in the regions of “poor” fit. Hence, the final energy grid calculated yielded electronic energies close to the quadrature points chosen by the HEG scheme. Table 1 gives the full discrete potential energy surface of LizK+. For rovibrational calculations it is important to obtain force fields that accurately interpolate the surface between calculated points. Various power series expansions [ 35-381 and PadC approximant expansions [ 35,391 have been used in rovibrational calculations. In particular, the embedded force fields using expansion variables of Dunham [ 361, Simon, Parr and Finlan [ 37 1, Ogilvie [ 3 8 ] as well as their exponential variants [ 35,391 have been successfully employed. Of the analytical representations investigated, a fifth-order Ogilvie power series expansion gives the “best” tit to the discrete surface. Using SVD it is possible to ensure that the analytical PE surface is free from singularities in the integration region [ 35 1. Table 2 gives the analytical representation of the PE surface of L&K+, which was used in subseto generate
250 Table 2 Fifth-order
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physics 172 (I 993) 24 7-258
Ogilvie analytical Expansion
representation variable
I Pl +Pz P3 P: +p: P: PIP2 P2Ps+PIP3 P: +p: P: PTP2 +P:PI PTP, +P:P, PIP: +P*P: P1P2P3 P: +p: P: P:Pz +P:PI P?P, +P:Ps (x2)“*=2.07x
b,
of the potential
energy surface of Li2K+ a)
Coefficient
Expansion
-613.94463 0.00000 0.00000 0.11496 0.18112 -0.03176 -0.00633 -0.16996 -0.19246 0.00198 -0.08911 -0.00164 0.16268 -0.24741 -0.00937 0.03978 0.31989
PIP: +P*P: P:P: P:P: +P:P: P:PzP3+PIP:P3 P*PzP: Pi’+P: Pi: PTP2 +P:Pr PYPS +P:P3 PIP: +PxPf: P’;Pi +P:P: P’:Pi +P:P: P:P: +p:p: PilPzP, +PIP:P, PIPZP: PTPZPI P:P*P: +P,P:P:
variable
b,
Coefficient -0.01953 -0.14775 0.04459 0.13761 0.19235 -0.15369 0.06355 -0.10163 0.17492 -0.13920 -0.05523 0.12259 0.78913 -0.05953 -0.03397 0.13696 -0.21469
lo-“
‘) SVD analysis is used [ 351 with singular values q4 and q6_32 being set to zero. b, Ogilvie expansion variable has the form 2 (R, - R,) / (R, + R,) , where R, and R, are the instantaneous respectively.
quent vibrational and rovibrational calculations. Furthermore, crZ4 and 0Z26_32singular values were zeroed, since these expansion terms unrealistically affect the magnitude of the high-order coefficients without significantly affecting the (x2)1/2 value. Fig. 1 gives the contour plots of the PE surface in terms of rectilinear vibrational coordinates (labelled as t coordinates [ 27 ] ).
3. Rovibrational Hamiltonian and solution algorithm The vibrational Hamiltonian used in this investigation is the Eckart-Watson Hamiltonian developed in t coordinate space by Carney et al. [ 271. The Hamiltonian has the form
and equilibrium
bond distances
where F,. is the vibrational kinetic energy operator, p, is the vibrational angular momentum operator, ow the Watson operator (which is a mass dependent contribution to the potential energy operator) and P is the electronic potential energy operator. The Watson operator is the sum of the diagonal elements of the reciprocal effective moment of the inertial tensor. A perturbation expression for this operator (which is valid only for small amplitudes of vibration) has been derived with respect to t coordinates [ 8 1. It was incorporated in the vibrational Hamiltonian, thereby avoiding the embedded singularities [ 8 1. The ab initio variational solution of the CzVvibrational Hamiltonian can be described as follows. Onedimensional wavefunctions are calculated using a finite-element solution of a one-dimensional Hamiltonian, which is expressed in terms of a single rectilinear coordinate and so incorporates only the firstorder expansion of the Watson operator. The firstorder expansion is diagonal in the t coordinates. For each t coordinate, 1000 finite-elements were constructed within the following domains: t, [ - 3.0,4.2], t2 [ -2.2, 4.81, t3 [ -4.0, 4.01. A three-dimensional
F. Wang and E.I. von Nagy-Felsobukl /Chemical Physm 172 (1993) 247-258
251
4, 35.
(4
3=-
@I
3-
3-
25.
25.
2. 2lS. :
15. l-
::
l055-
;
0.5,
I
O-
0.
. 05. . -1. -lS-
-05. -1. I . II
-2. b -25.
-3-5. -2. -25. 3.
3-
Fig. 1. Two-dimensional potential energy contour plots for the 5th order power series expansion using an Ogilvie expansion variable. The singular values uZ4and u26_S2are zeroed.
configuration basis was constructed from the three one-dimensional solutions [ 13 1. The configuration list was pruned by using a nodal cut-off criterion [ 15 ] ; that is, all products containing more than 13 nodes were excluded and so a total of 560 basis functions were employed. For the three-dimensional vibrational Hamiltonian, the third-order expansion of the Watson operator was used. All integrals were evalu-
ated using a sixteen point Gauss quadrature scheme, except the potential energy integrals which were evaluated using the HEG scheme [ 6-8 1. Finally, a secular determinant was constructed using eq. ( 1) and spanned by the configuration basis. It was diagonalised to yield vibrational wavefunctions and eigenenergies, both of which are required for the rovibration problem. Table 3 assigns the 10 lowest-lying
F. Wangand E.I. vonNagy-Felsobuki/Chemical PhysicsI72 (1993) 247-258
252
Table 3 Assigned vibrational band origins of Li*K+ (cm-‘)
000 100,010 001 200,110 101,011 002 010,100 300,210 201,111,101 002, 102
Symmetry
% Weight a)
Vibrational band origin b,
Al Al B1 Al BI A, Al Al B, Al
95 67, 18 91 44,20 57,14 82 71,22 27, 16 32, 13, 13 41,13
0.0 120.2 129.8 237.4 245.8 257.1 305.7 325.5 359.1 369.5
“~%Weight={C~/CC~}‘~*XlOO. b, The zero-point energy of Li2K+ 1s280.6 cm-‘.
vibrational band origins of L&K+. The rovibrational Hamiltonian in vibration trix representation is given by [ 27 ]
ma-
A~R=E,(S)~+O.~(A)~~~~+O.S(B),,~~ +o.5(c),,r41+o.5(D>,(14x14y+nyz7x) + (ilfi) (F)&,
(2)
where E, is the ith pure vibrational eigenenergy, (S) I, is the overlap vibration matrix element and l?‘s are the rotational angular-momentum operators, whose components refer to the molecule-fixed coordinate system. In general, matrix elements involving the rotational and centrifugal distortion operators are given
matrix elements spanned by lowest five vibrational eigenfunctions. The 0.5 factor in eq. (2) is incorporated in the matrix elements labelled AI-D’. The matrix elements spanned by plus and minus combinations of the regular symmetric top eigenfunction (RJ’km) have been detailed by Carney et al. [ 27 1. The advantage of this basis is that the corresponding matrix elements spanned by the angular momentum operators are real. Diagonalisation of the supermatrix HVR yields the rovibrational wavefunctions and eigenenergies. Table 5 gives the calculated rotational eigenenergies up to J= 3 for the low-lying vibrational states. In order to ensure convergence of the calculated eigenenergies with respect to the vibrational basis set used, a number of truncated basis sets were investigated. Calculations were employed using up to ten vibrational eigenfunctions and up to the same J level. It was found that using five and ten vibrational wavefunctions the mean difference for all the rotational levels is 0.001 cm-‘. The definition of the reduced Hamiltonians has been given by Watson et al. [ 401. Vibrational constants (such as the fundamental frequencies and anharmonic constants) could not be obtained within reasonable precision using a simple least-squares fitting procedure, due to the “mixed” nature of the configuration wavefunctions of the higher-lying states. Nevertheless, following the usual prescription [ 4 1,42 1, the spectroscopic constants were obtained frorn least-squares fits of rotational energy levels of the low-lying live vibrational states. The constants are given in table 6.
by
(C>,=(ilbzlj),
(W,=(ilPxyIj)
,
where i, j are the indices of the vibrational eigenfunctions and ,u is the instantaneous effective reciprocal inertia tensor. A Taylor series expansion is used for p [ 81 to circumvent the numerical singularities. The A-D matrices are symmetric with the (0) and (A) (B) matrix elements coupling adjacent odd or even values of K. The F matrix represents the Coriolis coupling that splits levels with non-zero K values (i.e. z axis angular momentum component ) in the presence of vibrational angular momentum. Table 4 gives rotational, centrifugal distortion and Coriolis coupling
4. Dipole moment surface Difficulties were encountered using the GAUSSIAN 88 package in order to generate a CI dipole moment surface #‘. Hence, the discrete dipole moment surface of the ground electronic state of L&K+ were performed at the Hartree-Fock SCF level. A 68 point dipole moment surface was generated in terms of the #’ In the August 1992 Bulletin of the Ohio Supercomputer Service, Dr. M. Frisch suggested to use the KEYWORD FORCE in GAUSSIAN 88 in order to obtam a CI dipole moment surface. However, due to limitations on our disk storage capacity this was not feasible using such a large basis set.
F. Wang and E.I. van Nag-yFelsobuki / Chemxal Physics I72 (I 993) 247-258 Table 4 Rotatronal
and Coriolis matrix elements
253
of Lr2K+ (cm-r)
Matrix element
A’
B’
C’
D’
F .’
11
0.6018 -0.004s 0.6016 0.0000 0.0000 0.6080 -0.0019 - 0.0065 0.0000 0.6016 0.0000 0.0000 -0.0054 0.0000 0.6084
0.1169 0.0072 0.1160 0.0000 0.0000 0.1158 0.0009 0.0101
0.0976 0.0049 0.0968 0.0000 0.0000 0.0965 0.0005 0.0069 0.0000 0.0960 0.0000 0.0000 0.0049 0.0000 0.0957
-0.4965x lo-“’ 0.9382x 10-r” -0.9002x lo-” 0.1445x 10-l -0.2474x 1O-2 -0.2608x lo-” 0.1597x10-‘0 0.1597x10-9 0.2073x lo-* -0.1360x 1O-9 0.1683x lo-* 0.1440x 10-l -0.1642x 1O-9 -0.4007x 1o-2 0.4788X 10-10
0.2702x lo-l9 -0.1480x 10-s 0.4339x lo-‘8 -0.6649x lo-’ -0.8586X 10-l 0.1928x lo-‘s -0.3184x 10-10 -0.3368x 1O-9 0.7090x 10-z 0.7363x lo-‘s -0.3010x10-* -0.1463x 10-l -0.13%3x10-9 -0.1089 0.1290x 10-l’
21 22 31 32 33 41 42 43 44 51 52 53 54 55
0.1151 0.0000 0.0000 0.007 1 0.0000 0.1148
‘) The F matrix elements have the relationship Table 5 Rotational
F( I, j) = - F(J, I).
energy levels of L&K+ for low-lying vibrational
J K.
10 11 11 20 21 21 22 22 30 31 31 32 32 33 33
Kc
1 1 0 2 2 1 1 0 3 3 2 2 1 1 0
states (cm-‘)
E” 0.0000
120.1477
129.8307
237.3582
245.8418
0.2145 0.6994 0.7187 0.6428 1.1090 1.1670 2.6217 2.6223 1.2839 1.7229 1.8390 3.2652 3.2679 5.7384 5.7385
0.2136 0.6992 0.7 176 0.6400 1.1080 1.1631 2.6202 2.6209 1.2786 1.7205 1.8308 3.2612 3.2637 5.7361 5.7362
0.2115 0.7038 0.7239 0.6338 1.1068 1.1670 2.6439 2.6445 1.2654 1.7107 1.8307 3.2780 3.2817 5.7906 5.7906
0.2124 0.6991 0.7166 0.6366 1.1066 1.1591 2.6191 2.6199 1.2723 1.7166 1.8219 3.2570 3.2593 5.7352 5.7358
0.2090 0.7027 0.7232 0.6261 1.1001 1.1616 2.6428 2.6436 1.249 1 1.6956 1.8178 3.2689 3.2735 5.7902 5.7903
rectilinear displacement coordinates #*. The grid used for the dipole moment surface differed from that used for the electronic energy calculations only with respect to its size (68 points instead of 73). Points located well away from the minimum energy geometry (i.e. rectilinear displacements larger than 2.0 au) were O2 Discrete authors.
dipole
moment
surface
available
on request
from
excluded in order to facilitate a more precise fit. The geometries were rotated and/or reflected at each data point in order to coincide with the Eckart frame, thereby that the dipole moment are computed with respect to the centre of mass. The molecule was placed in the xy plane, with the origin coinciding with the centre of mass and the positive x axis bisecting the included angle for the CzV symmetry.
F. Wang and E.I. von Nagy-Felsobukr /Chemical Physrcs 172 (1993) 247-258
254 Table 6 Spectroscopic
constants
of L&K+ for the lowest five vibrational
lko
v=l
Icar
-0.9234
A+C A-C rrsssan
0.2097x 0.1512x -0.0866 0.0792 -0.1823 -4.1984 -0.1031 4.5672
7cccc
rAAcc rX%cc 7AABB
reduction A, AJK AK 6, & first-order DJ DJK DK 6, & R6
states (MHz) v=2
-0.9240 10’ 105
0.2095x 10’ 0.1512~10~ 2.8446 -2.4594 2.8663 78.4762 1.2003 - 78.4480
constants 0.0009 -0.0692 0.1138 0.0101 -0.1251
v=3
-0.9245 0.2075 x 10’ 0.1578X105 -22.3850 -22.3176 - 17.7130 -138.2331 - 20.4923 558.0445
v=4
-0.9244 0.2090x 10’ 0.1510x10’ 15.2205 -4.4081 5.4620 149.3292 2.3855 - 146.2992
-0.9255 0.2006x lo5 o.1660x105 - 62.3548 - 70.9043 - 52.9245 -378.3192 -62.6884 1574.1887
distortion
centnfugal distortion -0.2848 1.6449 - 1.3146 0.0104 -0.2458 -0.1428
a) Ray’s asymmetry
-0.0481 -0.1627 -0.5057 -0.3315 -2.1921
5.5878 - 116.5932 115.433’7 0.0042 135.0186
-1.3515 2.7008 -2.7147 - 1.2268 -6.5668
16.6574 -333.2675 329.8412 -0.5343 380.4753
4.8669 -29.6529 24.0694 -0.3315 6.3996 2.4575
- 30.6869 101.0552 -65.9400 0.0042 - 106.6288 -18.1374
a. 1300 -54.1888 44.6932 - 1.2268 13.5095 4.7408
-85.8938 282.0394 - 182.9145 -0.5343 300.7253 -51.2756
constants
parameter.
For a charged triatomic molecule of CZVsymmetry, the p,, component at the equilibrium geometry is zero, whereas there is a permanent dipole moment for the pX component. A dipole moment surface can be generated using a power series expansion of the rectilinear coordinates [ 81. Table 7 gives the calculated regression coefficients of Li2K+ using the t coordinate expansion. From the coefftcients it is clear that the surface is dominated by the linear term (as the C, coefftcient is the largest). Nevertheless, the nonlinear terms are of significant magnitude suggesting that electrical anharmonicity cannot be neglected. The vibrational eigenenergies, wavefunctions and dipole moment surface were used in order to calculate the dipole moment matrix elements, Einstein transition probabilities (A,,,,, and II,,), band strengths (S,,,) and vibrational radiative lifetimes (7). Table 8 details these quantities together with the calculated transition frequencies for the low-lying vibrational states. Using the variational rovibrational wavefunctions, the dipole transition matrix elements are calculated
Table 7 Expansion
coeffkrents
for dipole momentum
surface of Li2K+ ‘)
Expansion variable
PX
Expansion variable
PY
G t1+ r2 (tt+f2Y t: (t,+t*)3 t:(t, +t21 (t, +tzY t: t:(tt +t*Y
0.83152 0.25788 0.08767 -0.02544 -0.03571 0.43869 -0.02987 0.07968 0.09000
ts f3(rt+f2) t: ts(f,+f2Y &t, +t21 t:(t, +r2) t3(ct+W t: f3(f,+f*Y
0.50276 0.47137 -0.00628 0.13353 -0.01253 -0.16115 0.00917 0.06001 -0.10591
0.01238 (ft+f2Y t%t, +tz1 -0.14193 t:(t, +t*Y -0.03616 p) All entries in atomic units.
&t* +f*Y
and utilised to calculate the transition matrix elements involving individual line intensities [ 43,441. Table 9 gives the variationally calculated rovibra-
F. Wang and E.I. von Nagy-Felsobukz / Chemrcal Physics I72 (I 993) 247-258
255
Table 8 Vibrational transition frequencies, square dipole moment matrix elements, Einstein coefficients ‘), band strengths ‘) and radiative lifetimes for Li2K+
i
1
1 0 20 30 40 5 0 6 0 70 80 9 0
V/l
fi;
4
4
41
(cm-‘)
(D2)
(s-l)
( 1016cm3erg-’ s-‘)
(atm-’ cm-*)
Tb) (s)
120.15 129.83 237.36 245.84 257.08 305.67 352.45 359.07 369.54
0.72x 1O-2 0.12 0.45x 1o-4 0.30x 1OP o.19x1o-3 0.26x 10-l 0.21 x 1o-4 0.70x 1o-5 0.57x 1o-4
0.0039 0.0799 0.0002 0.0001 0.0010 0.2311 0.0003
1.3584 21.9348 0.0085 0.0056 0.0352 4.8584 0.0039 0.0013 0.0107
9.6962 169.1836 0.1192 0.0815 0.5376 88.2262 0.0816 0.0280 0.2344
2.14 1.63 2.80 1.61 2.45 2.87 8.05 3.50 2.46
a) Calculated at 300 K. b, Lifetime of the upper state. Table 9 Variationally calculated rovibrational absorption line intensities (at 300 K) for X state of L&K+ VI J’ K:, K:
u” J”
K”P K”e
Branch P,
0 44 0 44 0 54 1 11 132 120 2 20 2 32 2 30 3 11 3 22 3 30 4 22 431 4 33
0 0 1 1 2 2 2 1 3 1 0 3 1 3 0
0 0 042 0 032 0 0 0 0 022 0 022 032 0 0
5 4
0 1
2
1
1 3 3 2
0 1 2 1
2
1
3 2
1 1
5 3 3 2 1 1 3 2 2 1 2 1 1 2 2
Q, R
-1 0 1 -1 0 -1 0 1 -1 0 1 -1 0
V4xB) (cm-‘)
.s,b) (atm-’ cm-‘)
Rk
‘)
6.86 7.32 7.01 119.74 120.14 120.57 128.74 129.85 129.99 235.44 238.87 236.01 245.22 245.70 250.52
0.210x 10-r 0.250~ 1O-2 0.265x lo-* 0.950x 10-3 0.933x 10-s 0.226x lo-’ 0.532x lo-* 0.141x10-’ 0.151x10-’ 0.180x 1O-4 0.184x 1O-4 0.180x 1O-4 0.125~ 1O-4 o.121x1o-4 0.131x10-4
0.378x 10’ 0.393x 10’ 0.458~ 10’ 0.717x lo-2 0.707x 10-Z 0.167~10-~ 0.355x 10-l 0.937x 10-l 0.988x 10-l 0.449x 1o-4 0.447x 1o-4 0.448x 1O-4 0.293x 1O-4 0.282x 1O-4 0.295x 1O-4
(Da)
‘) Difference of eigenenergies between rovibrational states labelled as A and X, where A and X represent the upper and lower rovibrational states. b, Band strengths between rovibrational states labelled as A and X, where A and X represent the upper and lower rovibrational states. ‘) Square of the dipole matrix element spanned by the rovibrational states are labelled as AX, where A and X represent the upper and lower rovibrational states.
tional
absorption
lected number
line intensities
of Li2K+ for a se-
of transitions.
5. Discussion A number of pseudopotential MO calculations have been performed on the electronic ground state of
L&K+ using small basis sets. Pseudopotential models do not take into account core-valence or core-core effects in a rigorous manner. On the other hand, we have used sizeable basis sets within the all-electron SDCI/FC ansatz. At small displacements from the equilibrium geometry, the single reference treatment is justified if the leading coefficients are large and nearly constant. This was shown to be the case for
256
F. Wang and E.I. van Nagy-Felsobuki /Chemical Physrcs 172 (1993) 247-258
alkaline-earth oxides, fluorides and hydroxides [ 45 1. On the other hand, the FC approximation is less satisfactory on theoretical grounds, even though it is tractable for more electron-dense systems. Nevertheless, comparison of the low-lying vibrational band origins of Li: using SDCI/Full with the SDCI/FC [ 6,7,19] gives some credence to this approximation at small rectilinear displacements. The SDCI/FC surface predicts Li2K+ minimum energy structure to be of CzV symmetry, with energy of -613.944625 E,,. The predicted equilibrium structural parameters ( RLIK, LiKLi bond angle) are (7.53 ao, 41 .SO), which compares well with pseudopotential-C1 value of (7.02 ao, 45.7” ) [ 5 1. Both calculations however differ more significantly with the Hartree-Fock pseudopotential value of (6.30 ao, 4 1.8’ ) [ 41. Similar differences between SDCI/FC [ 6-8 ] and pseudopotential-CI calculations [ 5 ] are shown in the structural parameters for the other Czy isovalent alkali metal cations. For example, for Li,Na+, LiNa: and K,Li+ absolute differences are (0.20 ao, 3.5”), (0.16 ao, 5.3”) and (0.23 ao, 6.7”) respectively [ 5-81. It should not be forgotten that core-core and core-valence interactions cannot be rigourously modelled in pseudopotential formalism [48-501 and so the total energy and therefore the predicted geometries are sensitive on the definition of the valence and core sub-spaces. The power series with the smallest x2 is not necessarily the “best” fit to a PE surface. It is important that the analytical representation is consistent with anticipated physical properties [ 12,13,18,35 ] and moreover, is smooth in the integration region, since numerical integration schemes are used to evaluate potential energy integrals. High-order power series are often problematic, because of singularities in the domain under consideration [ 12,35 1. In order to detect singularities, numerous graphical inspections need to be performed. Fig. 1 demonstrates that the analytical surface is smooth everywhere, with monotonically increasing repulsive walls within the integration region. The force field given in table 2 was used in the subsequent rovibrational calculations, since it provides the most accurate and smooth interpolation function. The fifth-order Ogilvie power series yields a (x2)l12 of 2.07x lo-‘, whereas for K,Li+ [S] a sixthorder exponential Dunham expansion variable was
employed yielding a (x2) ‘I2 of 2.57~ 10T4. On the other hand, for Li,Na+ and LiNa> [ 6 1, PadC force fields of order P (6,4 ) and P (4, 5 ) yielded the smallest (x2) ‘I*. Hence, the use of different expansion variables and power series of the force fields of these CzVisovalent molecules, suggests that there are subtle differences in their PE surfaces even close to the minimum. Table 3 gives the calculated vibrational band origins up to 370 cm-‘. As full mechanical anharmonicity is embedded in the vibrational Hamiltonian as well as operators coupling the t vibrational modes, the vibrational assignment is no longer simple, since mixing can occur for configurational basis functions belonging to the same irreducible representation. For example, configurational basis functions such as (000), (loo), (110) and (002) are all of Ai symmetry and so can have non-zero coefficients in the linear expansions of vibrational eigenfunction. Table 3 highlights this point by giving the percentage weight [ 6- 8,12,13,17-2 1 ] of the dominant configurational basis functions. It is clear that the harmonic approximation (within the framework of uncoupled modes), whilst intuitive, is too simplistic for interpretation of the dynamics of this molecule. The sequence of the five lowest-lying vibrational band origins of Li2K+ (given in table 3) is: (000)<(100)<(001)<(200)<(101).Thisisbasically consistent with the assignment for LiNa$ [ 6,7], although the fifth vibrational band origin of the latter is assigned as (0 10) and the ( 10 1) is assigned as the sixth. For Li,Na+ [6,7] the same sequence has the (0 10) vibrational band origin of lower energy to the (200) vibrational band origin, whereas for K,Li+ [ 8 ] the vibrational eigenfunctions are far more mixed, since configurational basis belonging to the same irreducible representation are nearer in energy and so can more strongly interact. It is anticipated that a larger configurational basis set expansion should not alter the sequence of the live lowestlying vibrational band origins, since for Li2K+, Li,Na+ [ 6,7] and LiNa: [ 6,7] the percentage weights of configurational basis functions in the vibrational eigenfunctions are large for most of these vibrational states. As expected, far greater mixing of the basis functions occurs for the vibrational eigenfunctions at higher energies and/or for the heavier isovalent molecules (such as for K2Li+ [ 8 ] ). Hence,
F. Wang and E.I. von Nagy-Felsobuki /Chemical Physics 172 (I 993) 247-258
larger configurational basis expansions may alter the sequence of these higher-lying vibrational band origins for all of the isovalent alkali metal cations [ 6-
81. The rotational, centrifugal distortion and Coriolis matrix elements spanned by the lowest five vibrational states are given in table 4. The rotational constants for the ground vibrational state are 0.602,O. 117 and 0.098 cm-’ respectively. The centrifugal distortion constants are expected to be small near the potential energy minimum and the calculations reflect this anticipation by yielding values of the order of lo-” cm- I. The largest centrifugal distortion constants are those spanned by wavefunctions belonging to different irreducible representations. These elements are of magnitude of 10-l to 10m2 cm-‘. Similar observations can be made for the Coriolis matrix elements. However, it should be noted that the Coriolis operator for an Eckart-Watson Hamiltonian yields numerically small diagonal matrix elements when compared with wavefunctions obtained from less diagonal Hamiltonian (s). The diagonal Coriolis elements are typically of order 1O-l9 to 1O- ” cm-‘, whereas the largest element is cs4, which has value of -0.1 cm-‘. For Li&+, the limiting case for the rotational levels is Mulliken’s prolate symmetric top, which is reflected by the calculated value of Ray’s asymmetry parameter of -0.92 for the lowest-lying five vibrational states. Hence, in table 5 the rotational energy levels up to the J= 3 level are assigned for the lowestlying five vibrational states, within the framework of a prolate symmetric top. Table 6 details the calculated rotational spectroscopic constants. obtained from our ab initio force fields and moreover, using our ab initio rovibrational states fitted to reduced Hamiltonians [ 40-421. High resolution rovibrational spectra are usually assigned in this manner, but using experimentally measured rovibrational levels. Although we have calculated the quartic centrifugal distortion constants at the equilibrium structure and neglected spin-rotation interactions, the signs and magnitude of these constants should be of assistance in the spectroscopic detection of Li*K+. In order to calculate transition probabilities, band strengths and vibrational radiative lifetimes a discrete dipole moment surface is required. Generally,
251
there are too few discrete dipole moment surfaces in the literature in order to estimate reliably the magnitude of error associated with a dipole hypersurface. Nevertheless, Green [ 491 has concluded that for a neutral diatomic molecule with a single sigma bond, using a double zeta basis set augmented with polarisation functions the error associated with a dipole moment at the Hartree-Fock limit is of the order 0.1 to 0.2 D. The basis set we have employed for Li2K+ certainly meets this criterion, although it is deficient with respect to producing a reliable Hartree-Fock limit. It should be noted that Li2K+ is bonded via single sigma bonds. However, what is not known is the error variation of the dipole moment hypersurface over small amplitudes of displacements. Table 7 indicates that the linear expansion variable is dominant, although the anharmonic terms are too sizable to neglect. Hence, care must be taken, since the error may not be uniform over the entire hypersurface nor of the order of 0.2 D. Table 8 gives the Einstein transition probabilities, band strengths and vibrational radiative lifetimes with respect to transitions from the ground vibrational state to the ninth excited vibrational state calculated using vibrational eigenfunctions and eigenenergies. For the CIV point groups there are no Raman forbidden transitions, unlike Liz which has D3h symmetry. Hence the lifetimes of the excited vibratibnal states of K2Li+ are small for these transitions compared with those of Li: [ 6,7,19,21]. In order to determine rotational line intensities the rotational partition function was obtained from the variationally calculated rotational levels. Neglecting contributions from the vibrational partition function, the intensities were calculated at 300 K. It would be anticipated that some rovibrational transitions should be accessible using laser spectroscopy, even though no such experimental data are currently available. Table 9 gives the absolute line intensities and the squares of the electric dipole transition matrix elements between the vibrational ground state and lowest-lying four excited vibrational states for some of the most intense transitions within the P, Q and R branches. It is hoped that these calculations will assist in the rovibrational spectroscopic detection of this molecule and moreover, will promote even more extensive configuration interaction calculations.
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Acknowledgement All calculations were performed using the VAX 3100 and VAX 665Os, the latter made available by the generous support of the Computing Centre, The University of Newcastle. We also wish to acknowledge the support of the Research Management Committee, The University of Newcastle. Finally, we wish to acknowledge the Overseas Postgraduate Research Award held by Ms. F. Wang.
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