J. Phys. Chem. 1992,96,9691-9696 of G. P. Diiges, H. Reinhard, and J. Ohlmann in providing for simulation programs is gratefully acknowledged. We thank Prof. Dr. L. Banci (Florence) for the gift of a sample of diethylenetriamineimidazole.
References and Notes (1) Solomon, E.I.; Penfield, K. W.; Wilcox, D. E. Srrucr. Bonding 1983, 53, 1. (2) Solomon, E. I.; Gewirth, A. A.; Westmoreland, T. D. In Aduanced EPR; Hoff, A. J., Ed.; Elsevier: Amsterdam, 1989; pp 865-908. (3) Mims, W. B.; Peisach, J. J . Chem. Phys. 1978, 69, 4921. (4) Van Camp, H. L.; Sands, R. H.; Fee, J. A. J . Chem. Phys. 1981, 75, 2098. (5) Iwaizumi, M.; Kudo, T.; Kita, S.Inorg. Chem. 1986, 25, 1546. (6) Henderson, T. A.; Hurst, G. C.; Kreilick, R. W. J . A m . Chem. SOC. 1985. 107. 7299. (7) Hkermann, J.; Kappl, R.; Banci, L.; Bertini, I. Biochim. Biophys. Acra 1988, 956, 173. (8) Van Camp, H. L.; Sands, R. H.; Fee, J. A. Biochim. Biophys. Acra 1982, 704, 75. (9) Yokoi, H. Biochem. Biophys. Res. Commun. 1982, 108, 1278. (IO) DBges, G. P.; Hiittermann, J. J . Phys. Chem. 1992, 96, 4787.
9691
(1 1) Hiittermann. J.; Kappl, R. In Metal ions in Biological Sysrem; Sigel, H.1 Ed.;Marcel Dekker: New York, 1987, Val. 22, pp 1-80. (12) Reinhard, H. Diploma-Thesis. University of Saarland, FRG, 1992. (13) Ovchinnikov, I. V.; Konstantinov, V. N. J . Magn. Reson. 1978, 32, 179. (14) Belford, R. L.; Duan, D. C. J . Magn. Reson. 1978, 29, 293. (15) Kevan, L.; Kispert, L. D. Elecrron Spin Double Resonance; Wiley: New York, 1976; p 18. (16) McFadden, D. L.; McPhail, A. T.; Garner, C. D.; Mabbs, F. E. J . Chem. Soc., Dalron Trans. 1976, 47. (17) Peisach, J.; Blumberg, W. E.Arch. Biochem. Biophys. 1974,165,691. (18) Sakaguchi, U.; Addison, A. W. J . Chem. SOC.,Dalron Trans. 1979, 600. (19) Basosi, R. J. Phys. Chem. 1988, 92, 992. (20) Yordanov, N. D.; Stankova, M.; Shopov, D. Chem. Phys. Lerr. 1976, 39, 174. (21) McGarvey, B. R. J . Phys. Chem. 1967, 71, 51. (22) Basosi, R.; Valensin, G.; Gaggelli, E.;Froncisz, W.; PasenkiewiczGierula, M.; Antholine, W. E.; Hyde, J. S . Inorg. Chem. 1986, 25, 3006. (23) Tainer, J. A.; Getzoff, E.D.; Beem, K. M.; Richardson, J. S.;Richardson, D. C. J. Mol. Biol. 1982, 160, 181. (24).Reinhard, H.; Kappl, R.; Hiittermann, J.; Bertini, I.; Banci, L.; Viezzoh, M. S. Manuscript in preparation.
Use of Rotated Jacobi Coordinates to Calculate Vibrational Levels of HCN Jo& Ziiiiiga,* Adolfo Bastida, and Albert0 Requena* Departamento de Qujmica Fisica, Universidad de Murcia, 301 00 Murcia, Spain (Received: April 1 , 1992)
A study is made of the use of rotated Jacobi coordinate systems to compute vibrational energy levels of HCN. The coordinate systems are derived by making an orthogonal rotation of the radial Jacobi coordinates, and the three possible arrangements of the Jacobi variables are comparatively considered. The vibrational self-consistent-field (VSCF) method is used to check the quality of the different coordinate systems. It is found that the Jacobi coordinates associated with the N-CH arrangement and rotated to transform into a system of curvilinear normal coordinates provide the best SCF energies and are the most useful to calculate the exact vibrational energy levels of HCN by the variational configuration interaction (CI) method.
I. Introduction In recent years, accurate theoretical determination of highly excited vibrational states of small polyatomics has become an area of central interest due essentially to the great advances in experimental techniques to probe such states.l-' In the high-energy regime, the vibrational motions are very anharmonic and coupled, and solution of the vibrational Schrodinger equation turns into a difficult task, even for triatomics. As a consequence, a variety of approximate and variational methods have been developed to deal with this problem?-'8 Part of this theoretical research has been focused on finding optimal sets of vibrational coordinates in which to represent the Hamiltonian and the wavefunctions. The reason for that is clear. Optimal coordinate systems will lead to small couplings between the vibrational modes, thus allowing not only to simplify the calculation of the vibrational states but also to obtain a better physical insight into the vibrational behavior of the molecule. In this context, different coordinate systems, including as examples local, Jacobi, hyperspherical, ellipsoidal, etc., have been studied, as well as techniques to improve their quality. 10~13,17-33 Jacobi coordinates and different variants of them have been successfully used to describe large-amplitudevibrational motions in triatomics along the last d e c a d ~ . ~ JThese ~ J ~ coordinates are naturally defined for treating atom-molecule collisions and their use is, in principle, well indicated for describing atom-diatom complexes and for isomerization ~ y s t e m s . ~Reently, J~ however, Jacobi coordinates have been shown to be also useful in typical bound state calculations.20~21~28~29~34~36 In fact, for triatomic molecules of C, and D3h symmetry and for those with linear equilibrium geometries, these coordinates can be transformed into
curvilinear normal coordinates by making a proper rotation of the Jacobi radial distance^.^^^^^.^^ Jacobi coordinates have also the advantageof being orthogonal; Le., the kinetic energy operator for them is diagonal. In this paper, we present a study of the quality of different rotated Jacobi coordinate systems to calculate highly excited vibrational energy levels of HCN. Concretely, we consider the systems associated with the three possible arrangements of the Jacobi variables. We usc the vibrational self-consistent-field (SCF') approa~h'~.~',~* to assess the suitability of the rotated systems in describing the vibrational motions of the molecule. We find that, for each Jacobi arrangement, the optimum value for the rotation angle is that for which the potential energy function is totally decoupled up to second order, so that the rotated system can be identified as a curvilinear normal system. Moreover, the best SCF energies are those obtained by using the rotated coordinate system corresponing to the N-CH Jacobi arrangement. Consequently, this coordinate system is shown to be clearly superior to the others in computing exact vibrational energies for HCN using the variational configuration interaction (CI) method. In section I1 we discuss the rotated Jacobi coordinate systems and the SCF and CI methods, and section I11 presents the results obtained and the analysis thereof. 11. Theory The internal Jacobi coordinates R , r, and 0 for a triatomic molecule A-BC are defined as the distance of atom A to the center of mass of BC (R), the length of BC diatom bond ( r ) , and the angle between rand R (e). The vibrational Hamiltonian in these coordinates can be written asz8
0022-3654/92/2096-969 1$03.00/0 0 1992 American Chemical Society
9692 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
ZliAiga et al.
c' a
where the reduced masses are
C
b
and the volume element is sin 8 dR dr de. By mass scaling the radial Jacobi coordinates R and r in the form S = dR (34 s = d-'r
(3b)
d = (CCI/PZ)~/~
(4)
with
c
the Hamiltonian becomes
where the reduced mass p is mAmBmC
.=(
1
(6) mA + mB + mc We can now define the rotated radial coordinates z1 and z2 as follows zI= S cos CY + s sin CY (7a) z2 = -S sin CY s cos CY (7b) where a is the rotation angle to be fixed. The Hamiltonian for the rotated Jacobi coordinates zl, z2, and 6 is then given byZ8
+
c
Figure 1. The three different sets of internal Jacobi coordinates for the HCN molecule corresponding to the H C N (a), N-CH (b), and C-NH (c) arrangements.
the H-CN arrangement has been used very recently by Gazdy and Bowman to calculate highly excited vibrational states of HCN.34*36 In order to check the quality of the different rotated Jacobi coordinate systems to compute the vibrational energy levels of HCN, we use the vibrational self-consistent-field method.13v37-38 The idea of optimizing coordinates by improving the vibrational SCF calculations was originally proposed by Truhlar and coL e f e b ~ r eand , ~ ~M ~ i s e y e vand ~ ~has been later applied to a number of problems including optimization of normal modes through linear transformati0ns~~-~~.29.M and comparison of different kinds of curvilinear coordinate systems.13~26~27~38~40 The theory of the SCF method has been preaented in detail in the l i t e r a t ~ r e ~ ~ q ~ ~ . ~ ~ and we give here only the relevant equations for the present problem. For the Hamiltonian (8), the SCF wavefunctions are given by where #n are the variationally optimized modes which +~~~n3(z19z2ie) = # n l ( ~ I ) ~ ~ ( 2 G2 n)3 ( @ )
are determined from the SCF equations where
For a general ABC triatomic molecule there are three different ways of defining the Jacobi coordinates depending on whether the r coordinate is associated with the AB, AC, or BC distances. Figure 1 displays the three sets of Jacobi coordinates for the HCN molecule. In the case of a symmetric bent AB2 system, the 0 mode for the arrangement A-B2 is separable from the r and R modes up to second order in the potential function.28 The 8 mode is therefore a curvilinear normal mode, in particular the mode corresponding to the asymmetric stretch.28 The r and R modes can be also transformed into normal modes by selecting the rotation angle CY as that for which the second-orderpotential coupling term involving zIand z2 is identically zero.*p For ABC molecules with a linear equilibrium geometry the above observations hold for the three Jacobi arrangements. Thus, for linear triatomics one can derive three possible sets of curvilinear normal modes from Jacobi coordinates. This is not, of course, the only way to obtain curvilinear normal-mode systems. Actually, they can be derived from any curdinear coordinate system by making the proper h e a r combinations; i.e., those for which the potential couplings up to second order are q u a l to zero.31*39However, the use of Jacobi coordinates as the starting point makes the required transformations very simple. In Table I we give the values of the rotation angles and the harmonic frequencies for HCN corresponding to the three Jacobi normal-mode systems. The one associated with
(10)
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9693
Vibrational Levels of HCN TABLE I: Harmonic Frequencies (in cm-') for HCN Corresponding
to Different Mass-scrled Jacobi Coordinate Systems" frequencies WS U S
*,I
% We
(CY
H-CN = -20.56O) 3316.2 2331.8 3450.7 2127.7 73 1.8
C-HN ( a = 30.70O)
(a = 8.78')
N-CH
2539.9 3159.7 2127.7 3450.7 731.9
2167.7 3425.7 2127.7 3450.7 731.7
~~
"The labeling A-BC means that the Jacobi coordinates are defined considering r as associated with the BC distance.
The SCF product form as eq 10 defines approximate wavefunctions. This approximation neglects instantaneous, as opposed to averaged, correlated motions in different modes. The SCF states form a complete set,however, and therefore the exact wavefunction for the multimode system can be expressed as an expansion in SCF states." Such as expansion, by analogy with the electronic structure case, is referred to as a configuration interaction (CI) wavef~nction.'~For the present problem, we can write for the exact eigenfunction $?I(zIJ~,@
=
E
6 , 1 n , n , ~ n ~ ( z l )4 n 2 ( z 2 )
4n,(@
(13)
n w l
The Schrodinger equation for $3zl,z2,B) yields the following matrix equation for the coefficients C,,,,,,
HC = ESC
TABLE 11: Comparison of Vibrational Energies (in cm-I) for HCN As Computed by the SCF Method Using Mfferent Rotated Jacobi Coordinate Svstems and bv the Variational DVR-DGB Method"
(14)
where E is the diagonal matrix of eigenvalues, S denotes the overlap integrals between the SCF basis states, H is the matrix which represents the Hamiltonian in the SCF basis, and C is the matrix containing the expansion coefficients. In order to circumvent computational complexities associated with the nonorthogonality of the SCF functions, we can use a set of virtual SCF-type states, each of which is a solution of the same SCF H a m i l t ~ n i a n . This ~ ~ set is orthogonal and its use simplifies the solution of the matrix equation (14). The converged CI results are, of course, numerically exact and independent of the coordinate system employed. The size of the basis set used in the CI expansion (13) is, however, expected to strongly depend on the coordinates used.
111. Numerical Results and Discussion We have used in the present calculations the empirical, frequency-optimized potential energy surface by Murrell, Carter, and Halonen (MCH)42for the HCN molecule. This surface gives a global description of all the regions of the potential, including the minimum corresponding to the HNC configuration and the isomerization barrier, and has been previously used by a number of authors to check different variational and approximate metho d ~ . ~ ~ In , ~this ~ paper * ~ ~we*focus ~ ~ our - ~attention ~ just in the vibrational states located around the HCN minimum. The computational implementation of the SCF method basically involves to solve the one-dimensional SCF eigenvalue equations and simultaneously to evaluate the coupling integrals appearing in the SCF equations. We use the discrete variable representation (DVR) technique15to accomplished both tasks since this method, which is based on the use of a pointwise representation as basis set to solve the eigenvalue equation, allows a very simple and fast evaluation of the Hamiltonian matrix elements. Solution of the SCF equations (1 1) using the DVR technique requires specification of the finite basis set to be used.@ We have chosen them to be harmonic oscillator functions for the stretches and Legendre polynomials for the bend. The harmonic functions are centered at the equilibrium values of the stretching zl and z2 coordinates with frequencies equal to those evaluated for the potential energy function (seeTable I). For each SCF state we use as many DVR functions as needed to reach convergence in the iterative SCF calculation. The SCF equations are solved as follows. For a given state labeled with the quantum numbers n,, n2,and n3, we first solve the bending SCF equation (1 IC) by approaching the unknowns SCF stretching modes &,(zI)and 4,,(z2) by the harmonic
n,, n2, n3b
C-NH
S C F energies H-CN
0, 0, 0
3562.20d 3505.74e 1614.12 1563.90 2560.76 2121.15 3253.28 3207.41 3098.60 33 12.92 4056.66 3653.55 4962.96 4920.30 5016.05 4 198.46 4572.99 4776.1 1 5667.23 5288.06
3527.76 3495.44 1522.14 1492.51 2337.18 2110.29 3042.13 3014.52 3219.25 3318.51 3792.52 3580.37 4589.95 4563.88 4606.74 4187.02 4647.09 4742.66 5294.00 5092.39
N-CH
Ease
3491.74 3485.23 1433.30 1427.00 2144.1 1 2098.77 2837.78 2831.58 3301.73 3323.15 3554.90 3510.72 4218.38 4212.21 4256.92 4174.19 4697.31 4718.37 4946.06 4902.40
3483.21 1418.33 2096.87 2806.54 3318.53 3508.60 4141.07 4172.85 4706.89 4890.58
"The ground state (0,0, 0) gives the zero-point energy and the rest of energies are with respect to the accurate ground state energy. bThe quantum numbers refer successively to the lowest frequency stretching mode (n,),the bending mode (n2),and the highest frequency stretching mode ( n J . cResults obtained using the DVR-DGB method (ref 43). dunrotated S C F energies. eRotated S C F energies.
finite basis set functions. Thus, we obtain a first approximation to the bending SCF mode &,(e), which is used to solve the stretching SCF equations (1 la) and (1 1b). The procedure is then repeated until convergence is reached for the total SCF energy. We note that the SCF eigenfunctions are not orthogonal and that the SCF levels for the excited states are not, in general, upper bounds to the exact values as in the case of the linear variational method, 13,22,51,52 In Table I1 we present the SCF energies obtained for the first several vibrational states of HCN using both the rotated and unrotated Jacobi coordinates systems for each arrangement, and the accurate variational results computed by Bacic and Light43 using the DVR-DGB method. The rotation angles are selected as those for which the rotated coordinates transform into normal coordinates. First, we see that for every arrangement the rotated SCF energies are more accurate than the unrotated ones for all levels considered. Moreover, the SCF energies optimize nearly around the rotation angles corresponding to the normal transformation, as it is shown for the H-CN arrangement in Figure 2, where the percentage errors in the SCF levels for the six lowest energy states with respect to the rotation angle are plotted. The differences between the rotated and unrotated SCF values are substantially large for the C-NH and H-CN arrangements, overall for states with some excitation in the stretching modes. This is expected of course, because the coordinatetransformation acts only on these modes. Comparatively, the bending overtone energies are less affected by the coordinate rotation. In second place, we observe from Table I1 that the SCF energies obtained by using the rotated N-CH coordinates are clearly superior to those from the rotated H-CN and C-NH ones. The improvements provided by the rotated N-CH coordinates are really large, mostly for t h e bending states for which the SCF energies from the other two rotated systems deteriorate very fast as excitation increases. The agreement between the N-CH rotated SCF energies and the exactum variational values is really quite good. Since the three rotated Jacobi systems are normal ones, Le., potential energy function for all of them is decoupled to second order, the better quality of the rotated N-CH system in describing the vibrational motions of the HCN molecule is expected to be due to a significant decrease in the magnitude of the higherthan-two potential couplings between the bending and the
9694 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
ZGlliga et al. TABLE I V Comparison of Excited V i b n t k d hergiea (in em-') for HCN As Computed by the SCF Metbod Using the N C H Rotated Jacobi Coordinate Systems and by the Variational DVR-DGB Metbod"
4 -
nl, n2, nsb 0, 0,0 1, 0,0 2, 0,0 3, 0,0 4, 0,0 5, 0, 0
(0,4.0)
0, 0, 0
0,0, 1 0. 0,2
-2
'
-30
J -24
-12
-18
-6
0
Rotation Angle (deg.)
Figure 2. Percentage errors in the SCF energy levels six lowest energy states of HCN as function of the rotation angle a using the H C N arrangement to define the rotated Jacobi coordinates (see text). TABLE III: Force Constants (in a J A-") for HCN O b t a i i Using tbe Three Different Nomial Jacobi Rotated Coordinate Svstem' k!jk
k2W k020
kW2 k3W
koso k210 k120
C-NH 3.3392 8.7834 0.4329 -4.2119 -24.4507 2.3705 -1 1.7004 0.3696 9.2263 3.3883 41.9840 2.4456 -4.0119 22.5783 7.3678 0.1125 -53.6766 -8.7751
H-CN
-
NCH -
-
0.1500 -
0.3296
-
-
-
-
-
-
2.02 1.90 1.33 0.3 1 -1.14 -3.03
3485.12 2098.77 4174.19 6228.18 8260.68 10271.65
EmC 3483.10 2096.87 4172.86 6227.87 8261.82 10274.68
3485.12 3323.15 6536.56 9648.45
3483.10 3318.37 653 1.02 9645.67
2.02 4.78 5.54 2.78
3485.12 1427.00 2831.58 42 12.2 1 5565.08
3483.10 1418.27 2806.43 4160.9 1 5477.32
2.02 8.73 25.15 51.30 87.76
5413.39 7482.17 9529.43 11555.08
5400.57 7462.01 9502.62 11522.48
12.82 20.16 26.81 32.60
5413.39 8620.21 11725.41
5400.57 8596.93 11694.12
12.82 23.28 31.29
3510.72 4902.40 6270.36 7610.95
3508.54 4890.48 6239.28 7550.74
2.18 11.92 3 1.08 60.21
4718.37 6093.26 7443.99 8766.48 3510.72 5572.75 7613.03 963 1.47 11628.01
4706.69 6064.82 7389.26 8675.70 3508.54 5577.20 7624.15 9649.3 1 11653.07
11.68 28.44 54.73 90.78 2.18 -4.45 -11.12 -17.84 -25.06
ESCF
ESCF
0.4098 4.4902
0.3025 -0.21 72
0.5278
-0.0008 -
4718.37 7901.80 10983.68
4706.69 7889.05 10973.56
11.68 12.75 10.12
-0.9627 -0.2550 0.8958
4718.37 6795.52 8850.88 10884.36
4706.69 6782.33 8836.48 10869.22
11.68 13.19 14.40 15.14
'Dashes indicate that the value of the force constant is the same as that for the C-NH rotated system.
3510.72 6795.52 9972.34
3508.54 6782.33 9948.81
2.18 13.19 23.53
stretching modes. In order to quantify the relative contributions of these couplings we have expanded the potential function in a Taylor series around the equilibrium point as follows
5413.39 6795.52 8 157.38 9495.22
5400.57 6782.33 8134.13 9452.64
12.82 13.19 23.25 42.58
695 1.19 8306.05 9633.92
6952.12 8294.57 9600.36
-0.96 11.48 33.56
9247.05 10568.19
9217.18 10512.14
29.87 56.05
k102 k012 k400 k040
kooa k310 k130
k220
k202 k022
k112
-
-0.6372 -25.2584 -4.1841
-
-
3 VQirQ2rQJ
= iC lk4
kijkaQ$Q!
(15)
where Q,and Q2are the displacement coordinates with respect to the equilibrium values for the lower and higher frequency stretching modes respectively, and Q3is the bending displacement coordinate given by r - 8. In Table I11 we show the force constants k,,k up to fourth order obtained using the three normal Jacobi rotated coordinate systems. It can be noted that the values of the force constants which couple the higher frequency stretching mode Q2 and the bending mode Qs (k012,k022, kI12)get considerably smaller when going from the C-NH to the N-CH rotated coordinate system, while not such large variations are observed for the coupling constants between the lower frequency stretching mode Q1and the bending mode (kIo2,k202).The normal rotated N
