Theory for the direct construction of diabatic states and application to the He+22Σ+g spectrum

Volume 192, number 5,6

CHEMICAL PHYSICS LETTERS

15 May 1992

Theory for the direct construction of diabatic states and application to the :E + spectrum C.A. Nicolaides, N.C. B a c a l i s a n d Y. K o m n i n o s Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48 Vas. Constantinou Avenue, 116 35 Athens, Greece

Received 24 January 1992; in final form 6 March 1992

We present a theory for the direct construction of correlated wavefunctions representing diabatic states. Attention is given to the proper separation and subsequent computation of state-specific, multiconfigurational, optimized wavefunctions whose main features do not change significantly with geometry and whose mixing causes the breakdown of diabaticity. In the case ofdiatomics, the theory is implemented via a method which obtains the zeroth-order description as a numerical MCHF function and the remaining "diabatic" correlation as an expansion in terms of numerical diatomic orbitals whose effective charges are optimized from the minimization of the energy. Transparent as well as accurate wavefunctions emerge. Application to the He~ 2Z~"valence and Rydberg states has produced results in agreement with those derived from knowledge of adiabatic curves and of d/dR matrix elements, obtained from extensive LCAO-MO configuration-interaction calculations.

1. Introduction Avoided or conical intersections o f molecular potential energy surfaces ( P E S ) o f states o f the same s y m m e t r y play a crucial r61e in a large n u m b e r o f physicochemical processes, e.g. refs. [ 1-3 ]. Much o f the discussion and analysis in this field uses the concept o f " d i a b a t i c " states, a n a m e coined by Lichten [4] in his pioneering analysis o f certain p r o t o t y p e collision phenomena. The "existence" or relevance o f diabatic states is related to the magnitude o f the f u n d a m e n t a l matrix elements which are present in the formal treatment o f nuclear m o t i o n [ 1-9 ]. Starting in the early 1960s, the related questions have focused on how to define them a n d / o r to obtain t h e m from first principles, since their wavefunctions are not eigenfunctions o f the frozen-nuclei electronic H a m i l t o n i a n ( a d i a b a t i c states), e.g. refs. [ 1 - 7 , 9 - 2 0 ] . To this end, one line o f research has concentrated on the c o m p u t a t i o n o f good adiabatic bases, o f n o n a d i a b a t i c or other types of matrix elements and o f suitable transformations which are expected to nullify or m i n i m i z e and smoothen the couplings due to nuclear m o t i o n while accounting for the effects o f interelectronic interac486

tions [ 1 - 3 , 5 - 7 , 9 - 2 4 ]. F o r quantitative applications on real systems, given the practical difficulties in computing and handling reliably the n o n a d i a b a t i c couplings, m e t h o d s utilizing correlated adiabatic bases while bypassing this requirement, seem p r o m ising, e.g. refs. [10,12,15,16]. In this Letter, we present results from an approach which aims at the direct calculation o f correlated ( q u a s i ) diabetic states, without first c o m p u t i n g or utilizing adiabatic states as bases. One advantage o f this a p p r o a c h is that it produces the valence diabatic states for very high energies, where the calculation o f adiabatic wavefunctions with the conventional LCAO-CI method is impossible. Application is made to the prototype He~- 2Z~- diabatic spectrum [ 1,4,17], where we have c o m p u t e d the diabatic PES and the state-specific correlated wavefunctions for the valence lt~,la~ state and the Rydberg l~g2nt~g, n = 2 , 3, 4, 5 series.

2. State-specific theory of diabatic states Let ~

represent a d i a b a t i c state labeled by n. In

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the present theory, the index n represents the statespecific zeroth-order representation of ~ and remains the same for all geometrical points G. Consider two diabatic states, 7'~ and ~ ' , with energies

15 May 1992

By definition, at points of intersection Go, they satisfy

tionary states [20] ~1 and the subsequent developments and applications of the state-specific theory (SST) of electronic structure [26]. For a fixed-nuclei spectrum, at each G there are adiabatic states, 7ti, for a particular symmetry, satisfying the electronic Hamiltonian eigenvalue equation

E~(Go) = E ~ ' ( G o ) .

Hel ~ i = E i T i •

E~'m(G) = (

~it/~'m(G)nel ~r~'m(G) )

.

( 1)

(2)

Although the energy of the diabatic states ~v~, ~ ' , etc. changes as a function of G, their overwhelming electronic structure characteristics are retained along the G line. On the other hand, those of the corresponding adiabatic states are not. As a consequence, around Go, integrals of differential operators, such as that representing radial coupling due to nuclear motion, have different features in the two cases. For the state-specific diabatic states the radial coupling is smooth as a function of G, whereas for the adiabatic ones it can become very sharp, or even infinite in the case of a conical intersection. The fact that ~ is not an eigenstate of H, led O'Malley [ 17 ] to treat these states as "quasistationary", and to apply an analysis similar to that applied to autoionizing states in terms of the formalism of projection operators. O'Malley implemented his theory on prototype systems in terms of simple valence bond functions with "projected atomic orbitals" [ 17 ]. In his summary, he concludes that his method produced diabatic states in half of the cases he examined. A many-electron approach to the computation of quasistationary states was published a couple of years later [20]. Although the immediate object was the formalism and practical computation of autoionization states, it was pointed out (ref. [20], footnote 73) that the same concepts and methods of statespecific computation of Hartree-Fock ( H F ) (or M C H F ) plus suitable projected correlation (rather than root-searching from matrix diagonalization) are also applicable to molecular quasistationary states. The importance of the dissociation region, where the adiabatic and diabatic states should be asymptotically equal, was stressed and it was suggested that previous difficulties [ 17 ] could be bypassed by the multiconfigurational representation of the possible near-degeneracies in ~P~.The framework of the present theory draws from this early work on quasista-

(3)

If ~ corresponds to a bound state, it is square integrable and E i is real. If it corresponds to an autoionizing state, it has outgoing-wave boundary conditions for the free-electron and E i is complex. We assume that at each G it is possible to compute a diabatic ~ with the following properties: (i) it shows only smooth configurational variation as a function of G, (ii) it has a PES which crosses that of other diabatic states at points Go where the adiabatic ~w avoid intersection, and (iii) it is assigned as a zeroth-order representation to a particular ~ . This leads to the possibility that one can always write a particular ~i in terms of diabatic functions ~ , ~ ' , etc. along the line G as follows: ~W= ~ + X~

(the correspondence

i~--,n depends on G ) ,

(4)

where X~ represents those parts of the function space which mix through H~I with ~ and destroy its diabatic character. Since there is no explicit Hamiltonian to which ~ is an eigenfunction, its construction is achievable in analogy with the concepts and methods of the SST of autoionizing states [20,25,26]. Thus, the above discussion leads to the required property that ( ~u~, 7'~ ) ~ m a x

(the correspondence

i~--~ndepends on G)

(5)

and that at the same G, the energy of ~u~ is minimum / ~ = ( ~ [Hel[ ~ ) = m i n ,

(6)

where E~ may be below or above E ~, depending on the geometry. In practice, eq. (5) constitutes a guideline for electronic structure considerations, while eq. (6) implies that ~ is suitably optimized #~ For a recent reviewon the theory and computation of nonstationary states see ref. [25 ]. 487

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for electron correlation corrections (see eq. ( 9 ) ) . The geometry dependence of eqs. (4) and ( 5 ) implies that in the region around Go the contributions from )(an to ~ increase while before and after the diabatic intersection, the index n is different for the same 7". On the other hand, in the dissociation region (symbolized by m ) which is physically observable and definable, Xa" tends to zero and ~i(oo) ~ ~g(oo) .

(7)

The physics of the dissociation region and relationships ( 4 ) - ( 7 ) allow the state-specific construction and computation of ~g. We write

~ = q)~ +x,q,

(8)

q) 3 is the single or multiconfigurational radially optimal Fermi-Sea [26 ] zeroth-order approximation, whose terms give ~ its reasonably constant major electronic structure features. The "diabatic correlation" X~ is that part of the electron correlation which contributes to the diabatic state without destroying its fundamental properties, which are to exclude the contribution of Xa~ to ~Vi while satisfying eqs. ( 1 ) (7). The zeroth-order description, qedo, is essential for the proper construction of ~ . Its accurate computation should be based on a state-specific HartreeFock or MCHF calculation which excludes configurations of Xg. Such configurations can be recognized a priori or during the computation. The higher the symmetry the easier it is for this desideratum to be accomplished. For example, for the case of diatomics which is examined here, the calculations of q) ~ are done numerically in spheroidal coordinates, and the character of the orbitals are recognized from their symmetry, quantum numbers and nodes. The diabatic correlation correction to q)3, X~, is written symbolically as

X ~ = ~ f f ~o + ~ f f 2 n + ....

(9)

where the single (o), pair (n), etc. correlation functions are expanded in terms of virtual orbitals which are coupled with the remaining orbitals in ~o, so as to form symmetry-adapted correlation configurations. These configurations exclude, by construction or imposed orthogonality contraints to core orbitals, configurations belonging to X~ of eq. (4), and are 488

15 May 1992

optimized by minimizing the variational energy from partial and total CI [20]. For both • ~ and X~ the dissociation region constitutes the initial conditions. From there, subsequent state-specific solutions ~u~ as a function of G are found by propagating into the interaction region along a chemically important coordinate, while keeping each state-specific • ~ in maximum overlap with that of the immediately previous geometry, starting with • ~(oo). Such a propagation can be carried to short distances where the crossing ~ reaches high energies for which methods depending on adiabatic wavefunctions break down.

2.1. Example: the NgH2 (Ng=He, Ne, Ar) ground and first excited singlet diabatic states In previous work [27], which dealt with the intersection of the ground and first excited singlet states of Hell2, Nell2 and ArH2 in Cs symmetry, these ideas were employed approximately in the following way. Firstly, two diabatic states, ~ ( o o ) and 7J~'(oo), were defined in terms of their LCAO-SCF functions, for N g + H z (~E + ) and N g + H ~ (B ~Eu+ ) dissociation channels. Since electron correlation was considered not to affect the physics at the crossing region, it was neglected, i.e. ~ - - , q~g, where q~o was a single configuration constructed in terms of state-specific SCF orbitals. For excited states, state-specific SCF orbitals provide immediately a much better representation of the true wavefunction than orbitals constructed from ground-state computations. Having obtained ~ (oo) and q~o ( ~ ) , the appropriate choice of the reaction coordinate was made by anticipating the appearance of the avoided intersection as a function of the covalent-ionic character of the H~ wavefunction. Thus, the computation was done in Cs symmetry as a function of the H - H distance ~2. The ~ and ~ differ by only one valence MO of the same symmetry, so that their separate computation in the LCAO-SCF approximation required special handling. Thus, at each point G, the solution was obtained while keeping a maximum ~2 The corresponding closely avoiding intersection adiabatic surfaces were first computed and interpreted in refs. [28,29]. More recently [ 30], very large computations indicated that for HeH~the adiabatic surfaces intersect.

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overlap with the SCF solution at the previous G. In this way, the closed-shell and the open-shell SCF solutions were obtained in an optimal way and indeed their crossing explained the quenching o f the H ~ B 1Eu+ -,H2 tZ~- radiative transition in the presence of He [27,31 ] and allowed the prediction of a similar phenomenon for Ne and Ar [27].

2.2. Nonorthonormality of molecular orbitals of different diabatic states Having computed the state-specific diabatic functions separately, their mixing at regions of intersection is done straightforwardly, by computing their interaction matrix element of the full-Hamiltonian. Because of their separate computation, the sets of one-electron molecular orbitals - computed in the LCAO scheme or numerically as in the present work (see section 3 ) - corresponding to each diabatic state are, in principle, nonorthonormal. Therefore, Nelectron integrals must be explicitly computed. This is feasible by implementing the method of corresponding orbitals [32], as has been done for correlated wavefunctions of the valence-bond type [ 33 ] or of the MO-CI type [16,34], although for the method of section 3 all molecular orbitals, zeroth-order and virtual, are computed numerically and not as LCAO.

3. Computation of the He + 21~+ diabatic spectrum

The H e r 2E~- diabatic valence state l~s, lcs~ 2E~-, has been the prototype of the notion o f diabatic states [1,4]. It crosses the Rydberg series lc2n~g (n = 2, 3...), and together they define a H e r 2E~- diabatic spectrum. Apart from the pioneering but approximate work of O'Malley [17] on the 1 6 , l t ~2 state, apparently the only construction of this diabatic spectrum where electron correlation is included, is that published b y Metropoulos et al. [35,36] using an indirect method, i.e. correlated adiabatic wavefunctions, calculations o f O/OR matrix elements and graphical interpolation. We have applied the theory of section 2 to this system as follows: Let v signify the configuration 1GglO"2 and n the configurations lO-g2n~g, where n = 2, 3 .... These configurations can be used to iden-

15 May 1992

tify the states for large R. The calculation of the diabatic correlated wavefunctions and energies corresponding to the states v and n is done directly, by applying the SST-based method for the computation of electronic structures of diatomic molecules [ 37 ]. In this approach, each state o f a given symmetry is treated separately under suitable orbital and configurational constraints pertinent to the problem under study, and not as a root of a diagonalized energy matrix on a basis of configurations constructed according to the LCAO-MO scheme. Specifically, each state of interest is assigned a multiconfigurational zeroth-order description, the Fermi-Sea function, which for atoms and diatomic molecules is computed numerically via the M C H F method [ 38 ]. In this way, inaccuracies in zeroth-order due to LCAO defficiencies are avoided, whilst for excited states, the radials o f the molecular orbitals are optimized automatically. Since excited-state wavefunctions are spacially extended, the numerical solution o f zeroth-order orbitals, which obtains automatically the long-range behavior, has a considerable advantage over the fixed-basis LCAO scheme *~3 Once the proper Fermi-Sea function has been obtained for each state o f interest, correlation configurations are added containing virtual orbitals (VOs) representing single, double, etc. excitations from the Fermi-Sea configurations. These molecular VOs are expressed as diatomic orbitals in spheroidal coordinates, whose numerical evaluation [41] is controlled by two parameters, the effective charges Z~, and Z~, that are optimized via the variational minimization of the total energy ~4 #3 In fact, from the study of atomic excited states, e.g. refs. [26,39,40] we know that very small, multiconfigurational functions whose orbitals are numerical and self-consistent, achieve the immediate description of the dynamics of interelectronic interactions much better than any large fixed basis set diagonalization method, including those using explicit ro dependence. #4 The efficiency and accuracy of this approach has been tested [37] on the prototype two-electron He2+ nE~ state, whose wavefunction contains a mixture of covalent and ionic terms over the potential energy surface (PES). For the SST calculation, 116 configurations are sufficient to produce lower energies over the entire PES (minimum, transition state and repulsive region) than those from the conventional LCAO-fulICI with 158 basis functions and 2282 configurations. 489

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According to the arguments o f section 2, the Fermi-Sea configurations for the v and n states representing the zeroth-order function ~o ofeq. (8) are v: log lo~, log2o 2, lo~2o~z, 2Og 1¢~o,2Cg 2 2 2Ou, log IOu2Ou, 2Og IO~20~, n:lo~nog,

(10a)

lo2unOg, n = 2 , 3, 4, 5 ,

(10b)

where for the calculation of ~o o f the valence diabatic state, the configurations o f n are excluded since they belong to the space Xa of eq. (4). • ~ and ~g are optimized self-consistently at each R. For example, at R = 2 . 0 au, the M C H F energy o f q~ is E~ = - 4 . 5 4 7 au (the H F energy o f the single configuration, l o g l o 2, is E ~ F = - - 4 . 5 2 5 au), while its main components are • ~ ~ 0.99( log l o 2) + 0 . 0 7 ( log I o~ 2Ou ) + 0 . 0 4 ( l o g 2 O 2)

(R=2.0au).

(11)

Similarly, at R = 2.0 au, the ~0 of the lowest Rydberg diabatic state is ~2 =0.93(lo22% ) -0.36(lo22%) (R=2.0 au),

(12)

with E~ = - 4 . 0 4 0 1 au. Having obtained the qb~ and qb ~, we add terms for the diabatic correlation, X~ and X,q (eq. ( 8 ) ) . For the valence state we consider only two VOs, the 3 ~ and the l n~ (the prime implies a virtual orbital) and construct 11 additional configurations, where the Iog orbital is never doubly occupied. Minimization of the resulting configuration-interaction energy

15 May 1992

yields at R = 2 . 0 au, E d = --4.5482 au but the level of purity remains approximately the same as in eq. (11). Similarly, for the n = 2 Rydberg state, only the singly excited configurations I o 2 3og, 1c 2 4og and 1of 5% were added as correlation vectors. The variational CI result at R = 2.0 au is E 2 = - 4.0402 au, i.e. the lowering is insignificant, while the main features of (12), which arise from the strong mixing l o ~ 1o2, remain. In fact, these features characterize the other Rydberg states as well, with almost identical coefficients. Thus, the character o f the diabatic Rydberg states n = 2 , 3, 4, 5 is given by the M C H F solutions R=l.4

~=0.99(lo2nOg)-O.17(loEnog),

R=2.0

~

R=3.0

~ =0.78( lo2nog)-O.62( loZnc~g) .

=0.93(lo2nog)-0.36(loEnog), (13)

The results of the computations are given in tables l, 2 and 3 and in fig. 1. Table 3 shows the energies and the nuclear distances at which the crossings occur, in the present calculations as well as in the previous ones [ 35] which, however, resorted to graphical interpolarion based on adiabatic functions and curves. The agreement is good.

4. Conclusion The direct construction o f diabatic states can be achieved if state-specific methods for the definition

Table 1 The state-specific energies and CI coefficientsof the main components for the diabatic valence state He~-, lo~ lou,22Zg+.The computation is explained in the text. For details of the method used for treating electronic structures of diatomic molecules with electron correlation, see ref. [37]. The configurations are: (1) loslo~, (2) log lou2ou, (3) log2o~z and (4) loB2~ R

1.00 1.20 1.60 2.00 2.40 2.80 3.00 490

E(HF)

- 3.0893 - 3.6259 -4.2358 -4.5251 -4.6671 -4.7393 -4.7614

E(MCHF)

-3.1345 -3.6633 -4.2640 -4.5477 -4.6904 -4.7625 -4.7851

E(CI)

-3.1372 -3.6660 -4.2672 -4.5482 -4.6938 -4.7661 -4.7887

CI expansion (1)

(2)

(3)

0.989 0.992 0.995 0.996 0.997 0.997 0.997

0.123 0.105 0.079 -0.070 -0.061 -0.060 -0.060

- 0.062 -0.053 -0.041 -0.035

(4)

-0.035 -0.033 -0.032

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15 May 1992

Table 2 The energies of the first four diabatic Rydberg states of He~, 1~ n o s 2Z~ (n = 2, 3, 4, 5 ) together with that of He 2+ 'Y~ (n = oo ) which was computed in ref. [37] R

n

1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0

2

3

4

5

-4.0812 -4.1355 -4.1255 -4.0949 -4.0634 -4.0402 -4.0295 -4.0651

-3.8007 -3.8699 -3.8725 -3.8535 -3.8322 -3.8176 -3.8216 -3.8584

-3.7027 -3.7758 -3.7827 -3.7667 -3.7480 -3.7358 -3.7443 -3.7831

-3.6579 -3.7360 - 3.7463 -3.7337 -3.7185 -3.7098 -3.7271 -3.7715

-3.5990 -3.6723 -3.6790 -3.6626 -3.6432 -3.6302 -3.6377 -3.6769

Table 3 The crossing points and corresponding energies of the H e r 2Z~ diabatic valence state lo, 1o 2 with the Rydberg lo2nos n = 2 , 3, 4 and 5 states, as well as with their limiting curve He 2+ ty~-. For states 2, 4 and 5, the agreement of the present computations (very small expansion) with the large LCAO-MO-type CI ones [ 35 ] is excellent. Around the energy of state n = 3, the MRD-CI calculations of Metropoulos el al. [ 35 ] gave two curves. Subsequent analysis has revealed that this is erroneous. The second state is of 2A symmetry n

E(R¢)

Re (crossing point )

2 3 4 5 He~ +

this work

ref. [ 35 ]

1.46 1.30 1.25 1.23 1.20

1.46 1.30, 1.27 1.24 1.22 1.20

ref. [ 17 ]

this work

ref. [ 35 ]

ref. [ 17 ]

1.2

-4.12 -3.88 -3.78 -3.74 - 3.67

-4.14 -3.89, -3.88 -3.79 -3.74 - 3.66

-3.6

?

~ __, ...J

"~-.....~

7... ,..

1.0

.1"

2.0

R (a.~)

2.5

3.0

Fig. 1. The part of the diabatic He + 2Z+ spectrum treated in this work as a function of internuclear distance R (all in au). The limiting curve of He~ + 'Z + which was computed by the same method [37] is also displayed. The valence curve l o g l a 2 (solid line) intersects the first four Rydberg curves l ~ n a g , n = 2 , 3, 4, 5 as well as the He~ + curve at the nuclear separations given in table 3. 491

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and computation of zeroth-order and correlated w a v e f u n c t i o n s are applied, as discussed in section 2. In c h o o s i n g the a p p r o p r i a t e z e r o t h - o r d e r description, the d i s s o c i a t i o n region offers itself for the chara c t e r i z a t i o n o f the d i a b a t i c w a v e f u n c t i o n . T h e state o f interest is t h e n f o l l o w e d i n t o the i n t e r a c t i o n region along a c h e m i c a l l y m e a n i n g f u l c o o r d i n a t e ~5, subject to c o n s t r a i n t s w h i c h secure the d e s i r e d configurational basic features o f the w a v e f u n c t i o n . T w o a p p l i c a t i o n s o f such ideas are p r e s e n t e d in section 2.1. a n d in ref. [ 16 ]. T h e q u a n t i t a t i v e a p p l i c a t i o n in this w o r k i n v o l v e d the efficient, state-specific calc u l a t i o n o f the H e ] zZ~- d i a b a t i c s p e c t r u m , w i t h w a v e f u n c t i o n s w h o s e z e r o t h - o r d e r q~0 (eq. ( 8 ) ) is c o m p u t e d n u m e r i c a l l y v i a the M C H F m e t h o d [ 38], while the r e m a i n i n g d i a b a t i c c o r r e l a t i o n is c o m p u t e d v a r i a t i o n a l l y by o p t i m i z i n g the e f f e c t i v e charges o f virtual n u m e r i c a l d i a t o m i c orbitals [ 37 ]. T h e w a v e f u n c t i o n features v a r y little o v e r the range o f i n t e r n u c l e a r d i s t a n c e s R = 1.0-3.0 au for the valence as well as for the R y d b e r g states, i n c l u d i n g the crossing points. ~s An analogous situation can be created with atoms, where "diabatic" states can be constructed as a function of the nuclear charge Z within isoelectronic sequences. For large Z, a particular valence-Rydberg pair is well-separated. For Z toward the neutral or negative ion end, the order may be reversed. If the wavefunctions of the two states are computed in a state-specific manner, their energies cross at some noninteger value of Z.

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