5098
J. Phys. Chem. 1982,86,5098-5105
Use of the State-Averaged MCSCF Procedure: Application to Radiative Transitions in MgO Randall N. Diffenderfert and David R. Yarkony' Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218 (Received: June 2, 1982; In Final Form: July 29, 1982)
A state-averaged multiconfiguration self-consistent-field(SA-MCSCF) procedure based on a density matrix driven MCSCF algorithm is presented and discussed in terms of its use at the MCSCF/configuration interaction (CI) level. The principal aspects of this study are the following: (a) SA-MCSCF optimized orbitals are used to approximate MCSCF orbitals. This aspect of the investigationis relevant to the not infrequent case where MCSCF wave functions are not attainable for higher roots of a given symmetry. (b) The dependence of extended CI wave functions on the choice of SA-MCSCF orbitals is considered and this property is used to provide an internal estimate of the reliability of interstate matrix elements. (c) Transition dipole matrix elements between the l l Z + and Z12+ states in MgO are determined. These results will be used to estimate line strengths for specific vibrational levels of the 2l2+ state. Also presented is a study of the convergence properties of two possible implementations of the SA-MCSCF procedure. It was found that, for a general averaging of several eigenstates, a procedure which includes second derivatives with respect to both orbital and configuration state function parameters is required.
I. Introduction In recent years advances in multiconfiguration selfconsistent-field (MCSCF) theory' have greatly enhanced the utility and scope of this method. Of particular significance here is the resulting ability to describe higher roots of a given symmetry a t the same level of accuracy that has traditionally been available for the corresponding lowest root. This MCSCF methodology has been used as the starting point in studies of the properties of polar molecules in general and the lighter homologues of the group 2A oxides, Be0 and MgO in particular.2 In these systems the desire to treat several electronic manifolds required the even-handed description of two or more states of same symmetry. The approach of ref 2 was to develop, from a large (e.g., full valence) MCSCF wave function, a compact configuration state function3 (CSF) expansion and a corresponding set of molecular orbitals optimized at the MCSCF level for each state in question. These results form the basis for a multireference single and double excitation configuration interaction3 (SDCI) calculation. The applicability of this approach, whose results were quite encouraging, is contingent on the determination of a compact MCSCF wave function as the zeroth-order description of the relevant state. This requirement may prove prohibitive, when, for example, variational collapse4 (root flipping) precludes optimization of orbitals for a compact CSF description of a higher root of a given symmetry. This appears to be the case in CaO where a preliminary treatment5 found that none of the 2,3'C+,23X+, and 233111states, all of which are of practical concern, are amenable to this approach owing to variational collapse. This problem, coupled with the need to treat radiative transitions in these systems, suggests the use of a stateaveraged MCSCF (SA-MCSCF) procedure to obtain the molecular orbitals corresponding to a limited CSF expansion. The SA-MCSCF approachs is frequently the method of choice in circumventing the nonorthogonality dilemma encountered in obtaining interstate matrix elements for states of the same symmetry and recently has become the subject of renewed i n t e r e ~ t .Roothaan ~~ has *Alfred P. Sloan Fellow. Present address: Department of Chemistry, Ohio State University, Columbus, OH 43210.
0022-3654/82/2086-5098$0 1.25/0
reviewed the ~ i t u a t i o n .Werner ~ and Meyer9 have discussed the use of this procedure in the construction of potential energy surfaces where the problem of variational collapse is clearly manifest. In this work we will investigate (i) the use of SA-MCSCF optimized CSF expansions to determine molecular orbitals in cases where individually optimized MCSCF wave functions are not attainable; and (ii) the dependence of extended CI wave functions on the choice of SA-MCSCF orbitals (4(w)see below) and the use of this dependence to provide an internal estimate of the reliability of interstate matrix elements. This study considers l'Z+(X) and 2lZ+(B) states of MgO. As indicated previously both the X and B states of MgO are amenable to description at the compact MCSCF level. This will enable a study of SA-MCSCF wave functions as a function of the weight vector, w (defined in eq A.l), including the limiting cases of first or second root only optimization. Since the states in question are well-characterizedexperimentally,'OJ1additional empirical standards of utility will be available. In this study a fully quadratic SA-MCSCF procedure based on the density matrix driven algorithm of Lengsfield12is used. The theoretical approach is outlined in section 11. A discussion of the methodological details of (1) M. Dupuis, Ed.,"Recent Developments and Applications of MCHF Methods", Proceedings No. 10, NRCC, Berkeley, CA, 1981. (2) C. W. Bauechlicher, B. H. Lengsfield, D. M. Silver, and D. R. Yarkony, J . Chem. Phys., 74, 2379 (1981), and references contained therein. (3) I. Shavitt, "Modern Theoretical Chemistry", Vol. 111, H. F. Schaefer, III, Ed., Plenum Press, New York, 1977. (4) A. Banerjee and F. Grein, J. Chem. Phys., 66,1044 (1977). (5) R. N. Diffenderfer, C. W. Bauschlicher, B. H. Lengsfield, and D. R. Yarkony, "Abstracts of American Conference on Theoretical Chemistry", University of Colorado, Boulder, 1981. (6) (a) J. Hinze, J. Chem. Phys., 59,6424 (1973); (b) K. Docken and J. Hinze, ibid.,47, 4928 (1972). (7) C. C. J. Roothaan and J. H. Detrich, and D. G. Hopper, Int. J . Quantum Chem., S13,93 (1979). (8)L. M. Cheung, T. S. Elbert, and K. Ruedenberg, Int. J. Quantum Chem., 16, 1069 (1979). (9) H.J. Wemer and W. Meyer, J. Chem. Phys., 74,5794,5802 (1981). (10) K.P.Huber and G. Herzberg, "Molecular Spectra and Molecular Structure", Van Nostrand-Reinhold, New York, 1979. (11)T. Ikeda, N. B. Wonr, D. 0. Harris, and R. W. Field, J . Mol. Spectrosc., 68,452 (1977). (12) B. H. Lengafield, J . Chem. Phys., 72, 382 (1980)
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 26, 7982 5099
State-Averaged MCSCF Procedure
the SA-MCSCF and CI procedures is reserved for the Appendix where, in addition, the convergence properties of two SA-MCSCF procedures are juxtaposed. In section 111the SA-MCSCF/CI results for MgO are presented. The w dependence of these results is discussed in terms of diagonal matrix element properties, the dipole moment ( p ) and total energy ( E ) ,and the derived spectroscopic constants, Re, T,,and w,. Analogous results for the interstate dipole operator matrix elements will permit assessment of the sensitivity of the interstate properties to the choice molecular orbital basis (@(w)). This represents a convenient ab initio estimate of the reliability of the matrix element in question, a useful result since total energies are frequently unreliable indicators of the accuracy of other matrix elements. A comparison with alternative approaches for treating higher roots is included. Section IV considers the chemical significance of the SA-MCSCF/CI results. The MCSCF wave functions are analyzed in order to provide a qualitative picture of the electronic structure of the l l Fand 2 l F states of MgO. A brief comparison with analogous results for CaO is included. The electronic transition moments are used to discuss radiative transitions between the X and B states in MgO. The results of this study are summarized and concluding remarks presented in section V.
11. Theoretical Approach The calculations reported in this work were performed at internuclear separations, R(MgO), between 3.0 and 3.7 (au) (atomic units) at intervals of 0.1 au. The extended bases of Slater-type orbitals (STO’s) of Clementi and Roetti13was used for both magnesium (7s5p2d) and oxygen (5s5p2d). This basis gives rise to 26 cr, 14 a, and 4 6 orbitals. ( a ) lZ+ States. The compact CSF expansion of ref 2 was used here to describe the llZ+ and 2l8+ states of MgO at the SA-MCSCF level. This expansion consists of the five lZ+ functions arising from the electron occupations (eo’s) (or electronic configurations) consisting of the fully occupied orbitals, lo-5a and la, coupled to the following open-shell structures 6a22a4
(2.la)
6 a7a2a4
(2.lb)
6u22a33a
(2.lc)
6u7cr2a33a
(2.ld)
The SA-MCSCF calculations were carried out on a subset of the grid (R(MgO),x) corresponding to the aforementioned values of R(Mg0) and the following values of x, the state averaging parameter: 1, 0.8, 0.5, 0.2, 0.1, 0.05, 0.002, and 0.0. For this two-state problem the SAMCSCF weight vector, W,which specifies the contribution of each state to the averaged energy functional, is given in terms of x by w = (x,l-x). (See part A of the Appendix for a detailed description of the SA-MCSCF procedure.) In the orbital optimizations only the cr and a functions were treated and, at the MCSCF stage, in which the fully second order procedure was used, the la(Mg,,) and 2cr(01,) orbitals were fixed at the single CSF SCF values corresponding to eo 2.lb. The SA-MCSCF results provide the basis for a description of these states at the extended CI level. The CI description, whose functional form is independent of both R(Mg0) and x, was determined in the following manner: (13) E.Clementi and C. Roetti, At. Data Nucl. Data Tables, 14,177 (1974).
TABLE I: Partitioning of Orbital Space classification sym-
metry
core
full
U
2
n
0 0
s
active
virtual
3
2
1 0
2 0
13 9 4
truncated 6 2 0
The orbital space, 0, was partitioned into core, full, active, virtual, and truncated subspaces as indicated in Table I. In each case (i.e., value of R(Mg0) and x) the full and active orbitals were those determined by the SA-MCSCF procedure. The unoccupied (virtual and truncated) orbitals were determined from the occupied (core, full, and active) orbitals by using the modified virtual orbital (MVO)14procedure for the Mg06+ion. The six u and two a orbitals with the largest orbital energies form the truncated subspace and were excluded from the CI calculations. In no case did this truncation eliminate an orbital with MVO orbital energy less than 10 hartrees or within a factor of eight of any retained orbital. The CI expansion was formed from the union of all CSF’s corresponding to (a) the reference occupations, (b) single excitations from a full or active orbital, and (c) double excitations from the active to the virtual space. This single and double excitation CI (SDCI) space includes 2247 electron occupations giving 7756 (Cmvsymmetry adapted) CSF’s.
111. State-Averaged MCSCF Methodology In this section SA-MCSCF/CI results for the l l Z + and 2lZ+ states of MgO are used to study several aspects of SA-MCSCF methodology. The diagonal matrix elements of H and p and the derived quantities Re, T,, and w e are used to consider the w dependence of P(w), for P an electronic property, with the aim of investigating the extent to which SA-MCSCF orbitals can be used to approximate (unattainable) individually optimized MCSCF orbitals. Interstate matrix elements of p are used to consider the possibility of using the w dependence of an interstate property as a measure of the reliability of that property at the level of CI in question. This latter point is of considerable practical import since total energies are frequently not reliable indicators of the accuracy of other matrix elements. Further, in the case of interstate properties for states of the same symmetry,the fact that ( 9$P1) # 0 when Q iand ql are described in terms of distinct, mutually nonorthogonal molecular orbital bases limits the applicability of methods based independently optimized states. ( a ) Diagonal Matrix Elements. Tables I1 and 111 present the results of SA-MCSCF and SDCI calculations, respectively. The monotonic dependence of E,(x)at both the SA-MCSCF and SDCI levels is gratifying. While this functionality is expected at the SA-MCSCF level it was necessary to include the single excitations from the full to the partial and virtual subspaces to obtain this monotonic dependence at the SDCI level. For R(Mg0) < 3.30 it was not possible to converge the SA-MCSCF procedure for w = (~,l-c)with c < 0.002. This problem is basis set dependent. It did not occur for two smaller bases, the basis set of ref 2 and a second basis of intermediate size composed of the 7s5p2d magnesium basis of Clementi and Raimondi15”and the double {oxygen basis of Clementi15baugmented with a 3d function (E = 1.39028) ~
_
_
_
_
~~
~
(14) C. W.Bauschlicher, J. Chem. Phys., 72,880 (1980). (15) (a) E. Clementi and D. L. Raimondi, J. Chem. Phys., 38, 2686 (1963). An additional 4p function (E = 1.0) was also included. (b) E. Clementi, ‘Tables of Atomic Functions”, IBM, San Jose, CA, 1965.
5100
The Journal of Physical Chemistry, Vol. 86, No. 26. 1982
'S
t-t-
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(9CO
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r
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i
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*
i
Diffenderfer and Yarkony
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$$ r-r-
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I
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N
i
i m r-Q,
ii ii 4i r - t - t-t- t-rN N
I
1
N N
I
t
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I
1
m a iri mt- m a N N
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i
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:
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N
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I
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i a
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I
'0
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i* iQ, d
State-Averaged MCSCF Procedure TABLE IV : MgO
The Journal of Physical Chemistry, Vol. 86, No. 26, 1982
5101
SDCI First Moments p ( I , J ) R (MgO )
I
J
Xb B
x
X X B
X
Bb B
X
X
B
X B X
X
1
B B
4
X B
X B B
1 19
X
X B
X
3.1
3.2
3.3
3.4
3.5
3.6
3.7
2.186 2.970 1.807
2.226 3.105 1.784
2.260 3.242 1.755
2.293 3.383 1.721
2.325 3.519 1.681
2.360 3.647 1.635
2.399 3.764 1.585
2.294 2.678 1.402
2.307 2.717 1.292
2.325 2.742 1.174
2.160 2.576 1.556
2.150 2.646 1.478
2.143 2.707 1.391
2.165 2.574 1.601
2.151 2.642 1.528
2.139 2.703 1.448
4 1
B B X
X B
a
1 0
X B
X
3.0 2.137 2.845 1.824
WC
1 1
B
X
2.296 3.234 1.735 2.182 2.453 1.806
2.221 2.512 1.766
2.264 2.526 1.688
2.278 2.562 1.595
2.283 2.631 1.504 2.193 2.545 1.540
2.158 2.227 1.832
2.176 2.223 1.785
2.181 2.342 1.738
2.178 2.419 1.685
2.169 2.499 1.625
1
2.177 2.499 1.664
499
B B
0 1
X B B
2.190 2.428 1.726
A t o m i c units used throughout.
X =
1'c state, B = 2'z+ state. +
and a 4p function ([ = 0.6) optimized to describe 0-at the MCSCF level. For R(Mg0) 1 3.30 the SA-MCSCF procedure is convergent for all x, enabling a comparison of lim, +(e,l-t) and +(O,l). From Tables I1 and 111it is clear that at both the MCSCF and SDCI bvel pi(O,l) - Ej(e,l-t)l 0 more rapidly for i = 2 than for i = 1. In fact the e = 0.002 MCSCF energies differ by less than 1 phartree from the e = 0.0 results for i = 2. A difference of 37 phartree is obtained at the SDCI level. Comparison of the o, and Re values from Table I1 shows that for x > 0.5 the potential energy function for the l'Z+ state is similar at the MCSCF and SA-MCSCF level. A similar trend is observed for the x < 0.5 results for the 2l2+ state. The difference between the x < 0.5 and x > 0.5 results is considered further below. Table IV presents the dipole moment matrix elements at the SDCI level, where similar trends are observed. Thus the diagonal matrix element results confirm the utility of the SA-MCSCF procedure for approximating (unattainable) MCSCF orbitals and also suggest the possibility of using an extrapolation procedure for defining the t 0 limit in cases where it cannot be obtained by direct means. SA-MCSCF is one of several approaches, including expanding the size of the MCSCF space: eliminatingx6or controlling degrees of freedom,l' and Hessian analysis,18 purposed to deal with the problem of variational collapse. It is of interest to consider the x dependence of Ei in Table I1 in the context of these alternative approaches. As indicated above as x 0, E2(x) E2(0)rapidly for x < 0.5. In this case section A of the Appendix shows that two conditions are simultaneously satisfied, the Hessian possesses one negative eigenvalue correspondingto \kl (see eq A.6), and \k2 satisfies the Hylleraas-Unheim theorem.lg These properties, which result from the structure of the
-
-
-
-
(16)C. W.Bauschlicher and D. R. Yarkony, J. Chem. Phys.,72,1138 (1980). (17)D. L.Yeager, P. Albertaen, and P. Jorgensen,J.Chem. Phys.,73, 2811 (1980). (18)J. Olsen, P.Jorgensen, and D. L. Yeager, J. Chem. Phys.,76,527 (1982). (19)(a) E.A. Hylleraas and B. Unheim, 2.Phys.,65,759(1930); (b) J. K.L. MacDonald, Phys. Rev., 43,830 (1933).
2.178 2.500 1.666
Unnormalized weight.
A subblock of the Hessian, are also found in the w = (0,l) result. The importance of the first of these properties has been considered by Olsen et al.18 For x > 0.5 the structure of the A subblock of the Hessian resembles that of the W = (1,O) case. The similarity of the x > 0.5 and x = 1.0 results for the first root was noted above. For x < 0.5, El is seen to degrade more rapidly than E2 improves. This suggests the existence of search modes (degrees of freedom) which have limited importance for the second root but degrade the lower root appreciably and ultimately result in variational collapse for R < 3.30. Table I1 illustrates the (strong) geometry dependence of this condition, compare x = 0.05 and x = 0.002 results at R = 3.20 and R = 3.30. The basis set dependence of this condition has been mentioned above. The effect of state averaging is to make searches along these baneful search modes unprofitable. Perhaps somewhat naively then, it performs a tacit mode analysis based on a variational criterion. Mode-explicit MCSCF procedures have been considered by Yeager et a1.l' (b)Interstate Matrix Elements. Table IV presents the transition dipole moments, p X , B ( x ) (the R(Mg0) dependence is suppressed), between the 112+and 2l2+ states on the (R(MgO),x) grid described above. As discussed previously, the use of a single set of orthonormal orbitals is necessary in this case to assure orthogonality of the CI wave functions. As in the case of the diagonal matrix elements, Table IV shows that the small x results approximate the x = 0 results quite well. Table I11 shows that even at the SDCI level the x < 0.2 (>0.8) orbitals do not provide a reasonable representation of the 1'2+(2l2+)state. This result is also evinced by the SDCI eigenvectors which exhibit appreciable contributions (a tolerance of 0.05 is used to define appreciable) from many single (and double) excitations. This is to be contrasted with the x = 1.0 and x = 0.0 (or e ) results for which the SDCI eigenvectors show no appreciable contribution from individual CSF's other than those in the MCSCF expansion. The x = 0.5 results occupy an intermediate ground in this regard. At the SDCI level there are no additional CSFs which contribute appreciably to the l'Z+ state and generally only two which contribute to the 2IZ+
5102
The Journal of Physical Chemistry, Vol. 86,No. 26, 1982
state. Thus the x = 0.5 orbitals should provide a reasonable basis for determining the 112+-212+transition moment at the SDCI level. This approach should provide a useful alternative to the orbital reexpansion technique frequently employed to evaluate interstate matrix eleIn this latter approach orbitals optimized for one state are used as the basis for a CSF expansion of the second state, i.e., the optimum orbitals for the second state are expanded in the orbital basis for the first state. This necessitates a larger CSF expansion for the later state which can be prohibitively large when the orbitals for the two states differ appreciably. Since the x = 0.5 orbitals are not optimal for either of the states in question the corresponding CSF expansions must be augmented with triple and higher excitations to reproduce the x = 1.0 or 0.0 results. A useful estimate of the expected magnitude of this effect on the transition moment can be determined from the x # 0.5 value of this matrix element. In this case the deemphasized state must also be augmented with triple and higher excitations. Those CSF's will improve the energy of the deemphasized root while contributing little to the transition moment (a one-electron matrix element). The magnitude of this effect (essentially a renormalization effect) is estimated, however approximately, by the change in the matrix element with w which for the case in question is approximately 10%.
IV. MgO Chemistry Spectroscopic constants have previously been determined for MgO at the MCSCF/SDCI level.2 Consequently the following analysis of the llZ+ and 2lZ+ energy matrix elements presented in section 111 is included only for completeness. The principal concern of this section is the analysis based on dipole matrix elements including dipole moments and radiative transitions, aspects of MgO chemistry which have received only limited theoretical attention.21 The results of Table 111, x = 1.0 for the l'Z+(X) state and x = 0.05 for the 2l2+(B)state were used to determine Re, o,,and T,with a three-point parabolic fit. The results for the X state are Re = 1.767 (1.749) A, we = 795 (785.1) cm-l and for the B state are Re = 1.728 (1.737) A, we = 796 (824.0) cm-'. T,is found to be 22289 (19984) cm-'. Here the experimentalloresults which differ only slightly from the deperturbed results" are given parenthetically. The dipole moments for the X and B states at the SDCI level are reported in Table IV (use x = 1.0 for the X state and x = 0.05 or 0.0 for the B state). The results for both states are consistent with a polar molecule, M$+O" with 6 N 2/3. This result shows that the simple valence bond picture of the B state arising as a result of a single charge-transfer excitation from the X state is somewhat naive. Tables V and VI consider the qualitative nature of the MCSCF wave functions for the l'Z+ and 2'2' states. Table V presents a Mulliken population analysis of the occupied valence orbitals and Table VI decomposes the wave function probability density in terms of the constituent eo's. The data are presented at R = 3.40 au which is typical of the near equilibrium results. Table V shows that the MCSCF orbitals are essentially the localized atomic orbitals expected in a simple valence bond picture so that in a qualitative sense each of the eo's in Table VI can be classified according to its value of 6, in Mg6+O*. (20)See, for example, S.R. Langhoff and J. 0.Amold, J. C h e n . Phys., 70, 852 (1979). (21)B. Huron, J. P. Malrieu, and P. Rancurel, Chem. Phys., 3, 277 (1974).
Diffenderfer and Yarkony
TABLE V : Mulliken Populationsa of Valence Orbitals for IX+ States at R = 3.40 stateb
2'r*
1lx+
0
Mg 60
70
s P d s
P d 2n
0.05
a
0.95
0.95
0.70
0.80 0.15
0.05
-0.05
0.35
s
1.0
P d
37r
0
Mg
1.0
s
p
0.10
0.80
d
0.05
0.05
Rounded t o nearest 0.05.
0.10
0.90
l l x +state
from w =
( 1 , O ) ; 2 ' x + state from w = ( 0 , l ) . TABLE VI: Probability Densitiesa by Electron OccuDationb for ' X + StatesC at R = 3.40 state
6022n4
60102n4
6022n33n
607u2n33n
l'X+
0.13 0.24
0.63 0.43
0.06 0.33
0.18 0.00
21x7
a R o u n d e d to nearest 0.01. xj=ikcz(j,i)f o r the k CSF's arising from the ith eo. 1 E + state from w = (1,Oj, 2'1;+ state f r o m w = ( 0 , l j .
The exception is the 3n function in the l'Z+ state which appears to play the role of a correlating function (see Table VI). For both the l l Z + and 2l2+ states the contribution from eo 2.la (6 = 2) is of secondary importance with eo 2.lb (6 = 1)making the largest contribution. Eo 2.lc (6 = 1)also makes an appreciable contribution to the 2l21+ state. This situation is to be contrasted with a similar analysis of the 112+and 2 ' F states in CaO presented elsewhere.** In CaO the analogue of eo 2.lc makes the largest contribution (with eo 2.la second) to the l l Z + state. However, in CaO the analogue of the 37r orbital (the 4a orbital) is much more covalent, being almost equal mixtures of oxygen p and calcium d functions. Thus the dipole moment functions for the l'Z+ states of MgO and CaO have distinctly different geometry dependences in the vicinity of their respective equilibrium geometries.22 The qualitative characterization of the 2lZ+ state of CaO is somewhat more involved and as mentioned above the details are presented elsewhere. We simply note here that the analogue of eo 2.lb makes the largest single contribution to the 2lZ+ state. However, in this case the analogue of the 7a orbital (the 9a orbital) which is again localized on the metal atom contains appreciable contributions from (calcium) 3d functions. The increased importance of d orbitals in CaO compared to MgO is consistent with the ionic character of these molecules and the stability of 2Dstate compared to 2Pstate of Ca+ as contrasted with the situation in Mg+ where the 2P state is more stable.23 Finally, we note that the preceding analysis of MgO is of qualitative value only, since as mentioned above Table IV shows the dipole moment function for both the 112+ and 2 l 2 + states is smaller than would be predicted by a simple ionic model, with Mg+O-. (22) R. N. Diffenderfer and D. R. Yarkony, J. Chem. Phys., in press. (23) C. E. Moore, Natl. Stand. Ref.Data Ser., Natl. Bur. Stand., No. 35 (1971).
The Journal of Physical Chemistry, Vol.
State-Averaged MCSCF Procedure
86,No. 26, 1982 5103
TABLE VII: SA-MCSCF Convergence Study QO E,"a'b
iteration
1
-274.392 -274.393 -274.393 -274,393 -274.393 -274.393 -274.394
2 3 4 5 6 12 a
In atomic units.
b
w = ( I / ~ ,l i s ,
QOC E,"a'b
AC
273 286 567 698 792 868 125 I/~).
0.636 8 9 4 0.102837 0.139 0 1 8 0.419 003 0.280 5 8 3 0.234 8 1 3 0.109 104
(1)
iij
(2) (3) (3) (3) (3)
273 873 609 315 346 346
0.241 3 0 2 ( 0 ) 0.230 7 8 9 ( o j 0.291 7 4 3 (1) 0.234 0 3 1 ( 2 ) 0.157 717 ( 5 ) 0.130 6 4 4 (11)
Negative of characteristic base 1 0 given parenthetically. c A = x ~ ( G ~- NQ - I ) ~ .
The transition moments, pBx(R), are presented in Table IV (use the x = 0.5 value). The results of this table are used to compute the A factors for the decay of the first vibrational transition, i f, f i n the X manifold, i in the B manifold, is given by
-
AB,X(i,f)= 2.0261 x lo4ai,fsB,x(i,f)
(4.1)
where aidis the frequency of the transition in cm-' and SB3(i,f) is the square of the vibrationally averaged transition moment expressed in au sB,X(i,f) = I(ilPB,X(R)lf)12
(4.2)
The vibrational wave functions required in eq 4.2 were determined by solving the vibrational Schroedinger equation with an RKR" curve for the X state and a Morse functionz5for the B state fit to the available spectral data from ref 10. Using this procedure the total A factor for transitions i = 0 and f arbitrary is A(O,*) = 4.2 X lo7 s-l. The discussion in section I11 suggests that the S factor leading to this result is an upper bound and should be reduced to approximately 85% of its calculated value giving A(O,*) = 3.6 X lo' s-l. The B state decays into both the A(1'II) and X states. The AB,A(O,*)factor for the B-A transition is not known precisely. Using pB-x/pB-A = 1.3 (for all R)" suggests a value of AB,x(0,*)/3.25is not unreasonable. This gives a lifetime (neglecting rotational effects-rotational line strength factors are summarized and standard spectroscopic notation and numerical conventions for transition moments given in ref 26) for the first vibrational level of the B state of approximately 21 ns. This value is in reasonable agreement with the available experimental data although the experimental value is far from ~ertain.l'.~~ A more detailed discussion of radiative properties of MgO and other alkaline earth oxides will be presented elsewhere.28
V. Summary and Conclusions In this work several aspects of state-averaged MCSCF methodology have been investigated, using a fully quadratic SA-MCSCF algorithm based on a density matrix driven MCSCF procedure.12 It was shown that for a general energy functional in which at least three states are averaged the fully quadratic procedure is necessary to obtain reasonable convergence. The SA-MCSCF wave functions for wi < 1were compared with the MCSCF results (wi = 1). For wi= 1 - t ( E small) the SA-MCSCF results approximate the MCSCF values quite well. This ~~
-274.392 -274.394 -274.399 -274.401 -274.401 -274.401
AC
~
(24)RKR curve communicated by R. W. Field. (25)P. M. Morae, Phys. Reu., 34, 57 (1929). (26)(a) E.E. Whiting, A. Schadee, J. B. Tatum, J. T. Hougen, and R. W. Nicholls, J.Mol. Spectrosc., 80,249(1980);(b) A. Schadee, J. Quant. Spectrosc. Radiat. Transfer, 19,451 (1978). (27)I. A. Svyatkin, L. A. Kuznetsova, Y. Y. Kuzyakov, and I. P. Leiko, Opt. Spectrosc., 48, 13 (1980). (28)R. N. Diffenderfer, D. R. Yarkony, and P. J. Dagdigian, J. Quant. Spectrosc. Radiat. Transfer, in press.
property of SA-MCSCF was used to extend the range of a previously proposed MCSCF/CI methodology2based on limited MCSCF expansionsto regions in which variational collapse precludes determination of MCSCF wave functions for higher roots. The SA-MCSCF results were used to obtain electronic dipole transition moments between the l'Z+(X) and 2lZ+(B)states of MgO. This analysis represents the first ab initio study of the R(Mg0) dependence of the B-X transition moment. The w = (0.5,0.5) results were shown to be optimal in providing an evenhanded starting point for characterization of those two states. It was suggested that x # 0.5 (w = x,l-x) results at the SDCI level provide an estimate of the renormalizationeffects on the transition moment which result from the x = 0.5 orbitals not being optimal for either state. The A factors for decay of the first vibrational level of the 2l2+ state of MgO into the 112+manifold were estimated with the ab initio transition moments and potential energy curves derived from spectral data. A predicted total vibrational lifetime for this level of 21 ns is in reasonable agreement with the available experimental data. Acknowledgment. The authors thank C. W. Bauschlicher (C.W.B.) and B. H. Lengsfield (B.H.L.) for helpful discussions and careful readings of early versions of this manuscript. The density matrix driven MCSCF program written by C.W.B. and B.H.L. which forms the basis for the SA-MCSCF algorithm used in this study was communicated by C.W.B. The authors are also grateful to R. W. Field for communicating his RKR results. The electronic integrals over Slater-type orbitals used in this study were computed on the Johns Hopkins University DEC-10 computer system with the ALCHEMY integrals program of B. Liu. The SA-MCSCF/CI calculations were performed on the CDC 7600 computer at the Ballistics Research Laboratory, Aberdeen, MD which was made available as part of a collaboration with G. F. Adams. This work was partially supported by grants from the National Science Foundation (CHE-7824153)and from the Air Force Office of Scientific Research (AFOSR-79-0073).
Appendix. Methodological Details A. State-Averaged Multiconfiguration Self-Consistent-Field Approximation. The SA-MCSCF procedure employed here is based on a density matrix driven MCSCF algorithm which treats all degrees of freedom to second order.12 For purposes of discussion our SA-MCSCF algorithm is summarized below. Each SA-MCSCF calculation is associated with an ordered K-tuple w = (w1,w2,...,wK) which denotes the weighted energy functional (Eav)
to be optimized. Here \kk is the kth eigenfunction of H in the space of the MCSCF expansion
5104
Diffenderfer and Yarkony
The Journal of Physical Chemistry, Vol. 86, No. 26, 7982
M *k
=
CCik$,
(A.2)
1=1
and E , is given by Ek = x t r pLJICIkClk
(A.3)
11
where p'sJ is the density matrix corresponding to basic CSF's $,, $, and I is the set of ordered (one- and twoelectron) integrals. Since (wI2= 1 and w,2 0, for K = 2 it is convenient to represent w = (cos28,sin28 ) by the single parameter x = cos2 8, 0 i x 5 1. Let the final orbitals cp and CSF expansion coefficients c be related to an initial set, 4', c', by # = #t' exp(-y) c = c' exp(-A) (-4.4) where A and y are general anti-Hermitian matrices. Since m,, = -mlJ 3 m, for m = A, y the upper triangle of m forms a vector (m) which enumerates the independent parameters of m . Requiring the first derivative of eq A.3 to vanish givesi2 (A.5)
where K
K
B = Ew,B,
Go = cw,G,'
fl=l
Amn,m'n'
fl=l
- -(En - E m ) ( w n - w m ) J m n , m , n
Here G,O and B, are the first and second derivatives of E, with respect t o y, w, = 0 for n > K, the 9,are required to satisfy Hm,, = 6,,nE, (see below) and pPi denotes the effect of the pth orbital mixing on the Hamiltonian matrix element between CSF's $, and In the above, y p ranges over the full upper triangle while A,,,, contributes only for the (2M - K - 1)K/2 values given by with n 5 K , m > n mn = m(m - 1 ) / 2 + n (A.7) The dimension of eq A.7 increases (rapidly for large CSF expansions) as a function of K in a procedure which includes both coefficient and orbital derivatives to second order (denoted QOC). It is independent of K when the procedure includes only orbital derivatives (denoted QO). For this reason feasibility studies for both QOC and QO procedures were performed. In our implementation, H is diagonalized at each iteration, and the augmented Hessian (AH) methodi2+Bis used to solve eq A.5. Since H i s diagonalized at each iteration, it follows from the last of eq A.6 that if A has J negative diagonal matrix elements, the J + 1 root of the AH is to be used to determine the orbital mixings in the QOC procedure. In this case the sign structure of the A matrix is analogous to that in an MCSCF procedure in which the J + 1 root is being optimized. The implications of this point were discussed in section 111. Note that for w,= wI a redundant variable corresponding to the mixing of roots i and j is eliminated from the calculation. It was found that the QO procedure is generally adequate if only two roots are averaged. However, for a three root SA-MCSCF, convergence of the QO procedure was too slow to be useful while the QOC procedure converges rapidly. This situation is illustrated by the following example. The basis set used is that described in section I1 and the (29) D. R. Yarkony, C h e n . Phjs. L e t t . , 77, 634 (1981)
CSF expansion which is indicated below is representative of a limited CSF description of the l1P,2lZ+, and 3IZ+ states in MgO (CaO). The expansion includes all l Z + CSF's arising from the election occupations: 6u22x4 6u7a2r4 6u22r33r 6u7u2a33x 6a8u2r4 6a8u2r33r 6a22r23r2 (A.8) 6a22r34a where the fully occupied 1u-5u, and 1r orbitals have been omitted for clarity. The starting orbitals were obtained from the single configuration SCF orbitals corresponding to the second electron occupation of (A.8) followed by six iterations of the QO procedure. This orbital conditioning procedure has proved effective in reducing the propensity for oscillatory behavior in the QOC approach.30 Table VI1 contains the results of this study. It is clear that in this case the QOC procedure converges rapidly while for the QO procedure convergence is too slow to be of practical use. B. Configuration Interaction Method. All extended CI calculations reported in this work were performed a t the multireference single and double excitation level in the full C,, symmetry. As discussed in section 111, eigenfunctions for a fixed CI expansion are required for a series of molecular orbital sets, #t(w). In axially symmetric points groups (with principal axis of order 2 3 ) the existence of twofold degenerate irreducible representations yields configurations (electron occupations) in which hundreds of Slater determinants are coupled to give tens of CSF's. These considerations have led us to develop a system of programs using a sorted formula tape m e t h ~ d o l o g yin~ ~ which the turnover r ~ l eappropriate ~ ~ t ~ ~the non-Abelian point groups and selection procedures involving incomplete electron occupations (e.g., an approximate% Hartree-Fock interacting space35or perturbation theory selections3 involving individual CSFs) are implemented simultaneously. For uniformity of treatment the H matrix formulas are expressed in terms of symmetry unique integrals over real functions. The incorporation of the turnover rule for non-Abelian point groups with an approximate Hartree-Fock interacting space and incomplete space type selection is outlined below, proceeded by a brief derivation of a turnover rule appropriate for the case considered herein. This aspect of the methodology can be incorporated into recently suggested direct symbolic CI procedures.36 The derivation, which is included to facilitate the subsequent discussion, closely parallels that of ref 32a. Let S be the invariant manifold of dimension N spanned by the Slater determinants Di,k = 1,N,corresponding to electron occupation (eo) i. Let P, be the projector corresponding to the invariant subspace of (spin-space) symmetry K of dimension M . The CSF's of symmetry K corresponding to eo i are given by N
Cc , , ~ ~ D ;
=
p=l
.
(B.la)
M
=
bD,xiSDi . .
(B.lb)
p=1
(30) C. W. Bauschlicher, P. S. Bagus, D. R. Yarkony, and B. H. Lengsfield, J. Chem. Phys., 74, 3965 (1981). (31) M. Yoshimine, J . Comput. Phys., 11, 449 (1973). (32) (a) E. R. Davidson, Int. J. Quant. Chem., 8, 83 (1974); (b) D. Munch and E. R. Davidson, J. Chem. Phys., 63, 980 (1975). (33) R. K. Nesbet, Ann. Phys. (N.Y.),3, 397 (1958). (34) W. C. Swope, H. F. Schaefer, and D. R. Yarkony, J . Chem. Phys., 73, 407 (1980). (35) A. D. McLean and B. Liu. J. Chem. Phvs., 58. 1066 (1973). (36) B. Liu and M. Yoshimine, J . Chem. Phis., 74, 612 (1981).
J. Phys. Chem. 1902, 8 6 , 5105-5110
Here Si = (S,’}p=lflis any M-dimensional linearly independent subset of the N functions of the form PJki, k = 1 8and the symmetry label K has been suppressed. In the following it is convenient to assume that the D i have been ordered so that Si= (PJki}k=lMNote that the c‘ and b’ depend only on the open shell structure of the eo i. Let X be any operator which commutes with the selfadjoint projector, P,. Since the \k,i are symmetry adapted according to irreducible representation K: .
N
.
(\k,’lXl\ki) = C cp, k and the new v(k) processed. Since each eo corresponds to an invariant subspace, the process must terminate, i.e., yield a linearly independent set, v(l),1 = 1,M thereby generating an appropriate ordering of the D i regardless of the method used to construct the CSF’s. It was shown34that for space-spin symmetrized CSF’s evaluated by a geneological coupling approach, an approximation to the Hartree-Fock interacting space35can be constructed by retaining only those CSF’s with the appropriate lineage. In this approach to CSF selection the original c‘ matrix may be retained and the CSF’s to be included in the CI expansion indicated by a “bit pattern”. In this case eq B.2b may still be used by simply failing to assemble the terms in that equation which correspond to unwanted matrix elements. A similar economy can be achieved in perturbation theory based selections3in which only some of the CSF’s corresponding to a given eo are retained.
Tunneling and Infrared Spectroscopic Characterization of Surface Reaction Products. 1. Chemisorption of Tetracyanoethylene on Partially Hydrated Alumina K. W. Hlpps’+ and Ursula Mazur Department of Chemistry and Chemical Physics Program, Washington State University, Pullman, Washington 99 164 (Received: June 7, 1982)
The reactive adsorption of tetracyanoethylene (TCNE) on two types of partially hydrated alumina is studied by vibrational spectroscopy. Reactive adsorption on thin-film alumina is analyzed with inelastic electron tunneling spectroscopy (IETS). These thin oxide films are grown on vapor-deposited thick films of aluminum by the action of an O2or H 2 0 vapor discharge. The formation of tricyanovinylalcoholate anion (TVA) on these thin films is demonstrated for the case of TCNE and tricyanochloroethylene(TCNCIE) adsorption from solution. Reaction of TCNE with A121s03in the presence of H2160vapor is shown to produce TVA(lBO).Solution-phase adsorption of TCNE on a high surface area y-alumina powder (bulk alumina) was also studied. IR spectroscopy on the adsorbed material proved of little value. Elution of the solid with acetone and subsequent chemical treatment of the filtrate provided a significant amount of CsTVA. Thus, the primary chemistries of thin-film and bulk alumina are identical in this case. Surface radical formation under these conditions amounted to about 1%of the total adsorbed TCNE.
Introduction During the past 15 years, there has been a considerable effort invested in understanding the reducing and oxidizing (redox) properties of metal oxides. Because many of these (alumina and magnesia for example) have a wide use as catalytic supports, the relevance of their redox properties in catalytic reactions has been extensively investigated. Most of the redox processes studied to date are those in
which an adsorbed neutral organic molecule transfers one electron to (or from) the surface of the oxide to become a cationic (or anionic) radical. The primary tools used in these studies are electron spin resonance and reflectance spectroscopy. It is believed that OH-, coordinatively unsaturated (defective) 02-,and 02-are the principal reducing agents on oxide surface~.~-~ The hydroxyl ion plays (1) Flockhart, B.D.; Scott,
Alfred P. Sloan Fellow.
T.A.; Pink, R. C. Trans. Faraday
1966, 62,730.
0022-3654/82/2086-5 105$01.25/0
0 1982 American Chemical Society
SOC.
