Faraday Symp. Chem. SOC.,1984 19 149-163 Studies of Molecular States using Spin-coupled Valence-bond Theory BY DAVIDL. COOPER Department of Theoretical Chemistry University of Oxford South Parks Road Oxford OX1 3TG AND JOSEPH GERRATT* Department of Theoretical Chemistry University of Bristol Bristol BS8 ITS AND MARIO RAIMONDI Istituto di Chimica Fisica University of Milan 19 via Golgi 20133 Milano Italy Received 3rd October 1984 The spin-coupled wavefunction for an N-electron system is described and the technical aspects (formation of density matrices energy minimisation generation of virtual orbitals) briefly outlined. The physical interpretation provided by this model is illustrated by calcula- tions on the potential-energy surface of the C++H system.The spin-coupled theory is quanti- tatively refined at the spin-coupled valence-bond stage. Results for calculations on CH+ LiHe+ and BH; are presented which show that at least for these systems chemical accuracy may be attained with 20&500 non-orthogonal 'spin-coupled structures ' (configuration state functions). Each eigenvector is dominated by one or two such structures thus providing visuality without sacrificing accuracy. 1. INTRODUCTION The basic idea of valence-bond (VB) theory is very simple. The wavefunction for the electrons in a molecule is constructed directly from the wavefunctions of the constituent atoms. This implements in a clear-cut way a large part of the experience of chemistry. VB theory provides a coherent explanation of a whole range of chemical phenomena the valency of atoms in different states the saturation of valency the directional properties of bonds and so on.Concepts such as avoided intersections between potential curves arising from 'zeroth-order ' states (e.g. ionic and covalent) form a basic part of our general mode of description of fundamental chemical processes.t This qualitative VB theory runs into difficulties as in the well known case of 0 and in the description of conjugated systems where simple molecular-orbital (MO) theory provides a more natural picture. Nevertheless it is clear the VB concepts are full of useful chemical and physical insights which exert a profound influence upon our view of molecular states and their evol~tion.~ However the translation of these attractive qualitative considerations into numerical results has always been disappointing.A very large number of terms is needed in the 7 A recent example is that of the Be molecule,' which possess an unexpectedly deep well of 714 cm-'. This can be understood as arising from the interaction between two structures the ground state (Be('S); Be(lS)} which gwes only repulsion and (Be(sp; 3P);Be(sp; 3P)>,which produces bonding.' 149 STUDIES OF MOLECULAR STATES wavefunction including many ionic structures whose presence in pre-eminently covalent situations is to say the least unexpected. The reason for this is clear. On molecule formation the participating atomic states are deformed. This deformation is expressed by means of ionic or charge-transfer structures as was first demonstrated by Coulson and Fi~cher.~ Consequently as long as one uses linear combinations of undeformed atomic states to describe molecular states convergence of the wavefunction will be very slow.These shortcomings of the classical VB theory are overcome by the spin-coupled VB theory at least for small systems. The spin-coupled wavefunction is of VB type but uses optimised orbitals whose forms vary with nuclear geometry. The wavefunction and its physical interpretation are described in section 2 together with an outline of the technical implementation of the theory. We illustrate the concepts which emerge from this model in a recent study of parts of the potential surfaces of the C++H system.The spin-coupled wavefunction is quantitatively refined by the addition of excited spin-coupled structures to form the spin-coupled VB wavefunction. This extension of the theory is described in section 3. Finally in section 4 the spin-coupled VB theory is applied to a study of the lC+ states of CH+ the lZ+states of LiHe+ and the potential surfaces of the B+ +H system. We show that at least for these systems the present modern form of VB theory is capable of providing results of the same accuracy as the highest quality MCSCF-CI wavefunctions while retaining an essential chemical and physical visuality. 2. THE SPIN-COUPLED WAVEFUNCTIONS5* In order to retain the fundamental physical content of the VB approach while allowing for a compact description of the deformation of atomic states we construct the following wavefunction for a system of N electrons ySM = cSkdN! d{41(1)42(2) **.4N(N)@F~M;L) k {#I 42 ...#N)-The orbitals #,(i) = #,(ri) are a set of Ndistinct non-orthogonal spatial orbitals. They are represented as quite general linear combinations of basis functions m 4 = c cppxp p=1 where m is the number of such basis functions much as in MO theory. The coefficients cppare determined simultaneously with the spin-coupling coefficients CSk by a second-order energy-minimisation procedure described below. The @$ k are a set of orthonormal N-electron spin functions which are eigen- functions of S2and S,with eigenvalues Sand M,respectively as shown. The index k denotes the specific mode of coupling the individual electron spins to form the final resultant spin S,and wavefunction (2.1) is a linear combination of all the modes of coupling that are relevant for a particular problem.The spin functions are usually constructed according to the branching diagram as described in ref. (5) and (6). However an alternative basis may be used if it makes physical sense to do so. The operator d is the usual N-electron antisymmetrising operator. Wavefunction (2.1) is extremely flexible. The practical implementation of the model represented by it rests upon a number of technical developments which we briefly D. L. COOPER J. GERRATT AND M. RAIMONDI 151 describe below. Further details may be found in ref. (5) and (6). The total energy corresponding to YsMis given by The operators hand g are the usual 1-and 2-electron operators and A D(pI v) and D(pvI az) are respectively the normalisation integral and 1-and 2-electron density matrices.These are connected by relations of the form N A = c D(Plv)<4pI4v> v=1 N DblP2 I Vl v2) = "3 z =1 mu,P2 P3I Vl v2 v,) (#p3I 4J N DOIl...pN-lIV1...VN-l)= x D(~~.*.~N-l~~~vl"'vN-lVN)(~p~I~V~)' vN=l It is understood that no index on a given side of the vertical bar in D(... 1 ...) is repeated. Eqn (2.4) provide the basis for the simultaneous computation of all the necessary density matrices by an extremely fast recurrence technique. We begin from the N-electron density matrix D(pl...pN 1 v ...vN) which is a list of purely group- theoretical quantities,8 and form directly from it the 4-electron density matrix.From this we form in succession the 3- 2- and 1-electron density matrices and A as specified in eqn (2.4). This procedure is very convenient since all of these quantities are required in the calculation. Implementation of this recurrence method has led to the development of very efficient methods of list processing which are well adapted to the newer generation of machines with large virtual memory. If we write the set of coefficients (cpp,cSk)as a single vector c the total energy E is minimised by the 'stabilised Newton-Raphson' (C+a1)6c= -g. (2.5) Here g is the vector of gradients of E with respect to the parameters cp and CSk Gis the matrix of second derivatives of EAq with respect to the same parameters and 6cis the vector of corrections to c.The orbital part of the gradient is given by N z cv4"-&p4p (P = 1,2 ...,N 1. (2.6) v=1 The operators RpVcontain 1- 2-and 3-electron density matrices and the parameter cp is interpreted as the average energy of orbital 4p in the presence of the other electrons. The second-derivative matrix possesses the following structure The diagonal block G(p,v) represents the second derivative of E with respect to the coefficients cpp,cvqoccurring in the orbitals 4 and d,, and contains I- 2- 3-and STUDIES OF MOLECULAR STATES 4-electron density matrices. Block G(c, c,) represents the second derivatives of E with respect to the spin-coupling coefficients CSk,and the off-diagonal blocks C@,c,) G+@,c,) contain the cross-derivatives of E with respect to orbital and spin-coupling coefficients.The parameter ain eqn (2.5) is given by a= -e0+0.5(gIg) where e is the lowest eigenvalue of G (assumed negative) and (g Ig) is the length of the gradient vector. As soon as e becomes positive a is put to zero and the procedure becomes the pure Newton-Raphson method. We have now accumulated much experience with the stabilised Newton-Raphson procedure. Our usual strategy is to begin a calculation at a point on the surface where all the atoms are well separated. Using an intuitive starting guess for c convergence is attained in 10-15 iterations. All coefficients are refined to at least i.e.the energy is correct to Convergence at neighbouring points is then reached in 5 or 6 iterations.Local minima are found occasionally but in all cases so far the corresponding wavefunction turns out to possess a clear-cut physical interpretation usually that of a nearby excited state. The orbitals 4pare almost always semi-localised the extent of the deviation from pure atomic form varying with nuclear geometry. As the interatomic separations increase the distortions diminish so that at sufficiently long range and provided that the correct spin couplings have been included the spin-coupled function (2.1) becomes identical to an antisymmetrised product of undeformed atomic states. The physical significance of the spin-coupling coefficients CSk is that they allow for a progressive recoupling of electron spins as the interacting atoms approach.At large interatomic separations the CSk coefficients assume values characteristic of the mode of coupling of the spins in each isolated fragment. At shorter ranges the coupling changes over to one which typifies the newly formed molecule. In the case of a chemical reaction the spin coupling changes from one characteristic of reagents to a coupling associated first with an intermediate (if there is such) and then with products. These variations in the spin-coupling coefficients are accompanied by corresponding changes in the orbitals. The chief qualitative finding to emerge from our work so far is that in nearly all cases the spin recoupling and orbital deformations occur quite suddenly over a narrow range of internuclear distances frequently between ca.4.50 and 5.5a0. Beyond the ‘critical radius’ of ca. 5.5a0 the system essentially consists of reagent atoms and molecules. At distances < ca. 4.5a0 both the orbitals and spin couplings are substantially characteristic of the product molecule or of the intermediate. This phenomenon is clearly shown in a recent study of the reactionlo C++H,-,CHl-+CH++H. (2.9) The mechanism and rate of formation of CH; from C+ ions and H molecules in the gas phase at very low temperatures are of great significance in interstellar chemistry.ll The overall reaction however is endoergic. Sections of the potential-energy surfaces have been previously calculated by Liskow et a2.12 and Pearson and RoueF3 using the MO-CI method. Extensive MCSCF-CI calculations have also been carried out more recently by Sakai et aZ.14Spin-coupled and spin-coupled VB studies of this system are in progress and we here consider the perpendicular approach of C+(,P) to H,.The basis set consists of triple-zeta (1s 12s I2p) functions on C+ and double-zeta (1s 12s I2p) STO on hydrogen atoms H and H,. This should be sufficient to account for the polarisation of H by C+. D. L. COOPER J. GERRATT AND M. RAIMONDI tH’ b H* Fig. 1. Coordinate system for C+ +H,. The coordinate system employed is shown in fig. 1. Ignoring the two core electrons on C+ the spin-coupled wavefunctions are of the form WCW) =@lo2 03 04 451 (2.10) Orbitals o and o2are deformed ls(H) functions and o3and o4are in general two hybridised C+ orbitals whose typical form is (2.11) At short interatomic separations orbital o3 consequently possesses a maximum amplitude along the C+-H direction and o4similarly along C+-H,.The two orbitals are interchanged by the operation of reflection in a plane passing through the C axis perpendicular to the molecular plane. Orbitals o1and 0,are similarly reflected into each other by the same operation. As R becomes large orbital 45assumes one of the following forms (2.12) In the present case of N =5 S =$ there are a total of five spin couplings corresponding to the paths on the branching diagram shown in fig. 2. With the orbital symmetries as above the coupling k = 3 is forbidden for all of the lowest energy states of eqn (2.10). In addition the coefficients c; and ci2 corresponding to the spin couplings k = 1 and k =2 must be related by cil =(1/2/2) ci,.The fulfilment of these symmetry constraints serves as a useful check on the operation of the programs. Cuts through the potential-energy surface for various values of R for the lowest ,B state are shown in fig. 3.15 The minimum-energy path runs in a deep entrance valley almost parallel to R through a long-range attractive well with R remaining close to 1.4uo the equilibrium separation between H and H in the isolated H molecule. At R = 1.85aothe path runs into a steep repulsive wall and turns sharply becoming almost parallel to R and climbs over the energy barrier. Thus as R decreases by just 0.05~~ to 1.80a0the value of R for the minimum increases to 3.80a0 The C+ ion suddenly inserts itself into the H1-H2 bond.Once over the barrier the path plunges directly down into the minimum in the plane which occurs at R =0. Associated with this is a complete change in the spin couplings as shown in table 1. Before the repulsive wall at R = 1.85a0 is reached all spin-coupling coefficients are essentially zero except ci5 which corresponds to the reagents C+(,PP)and H,(lZ;). STUDIES OF MOLECULAR STATES 312 1 1 /2 LLl 12345 so 3/21 (C) 112 'Lu!l 012345 w N Fig. 2. Branching diagrams for N =5 S =a:(a) c, (b)c, (c) c3 (d) c and (e) c,. -3832 ,,I,(,II ,,,,I,,1 ,(,,,,,,,,,,,,,,,,,,,11 1 2 3 L 5 1 2 3 L 5 -3832 -mrr-38487 I I I I I I I I I ,I11 I 11 I I II I I I I I I 1 2 3 L 5 1 2 3 4 5 H-H internuclear distance (a.u.) Fig.3. Potential-energy surfaces C++H, ,B for various values of R, (a) 2.50 (b) 2.29 (c)2.20 and (d) 1.80. D. L. COOPER J. GERRATT AND M. RAIMONDI Table 1. CH;(2B1) minimum-energy path in the region of the energy barriera E Cf1 Cf3 1.90 1.60 -38.359 330 0.022 678 0.032 071 0 0.137 821 0.989 678 1.87 1.60 -38.351 617 0.024 042 0.034 001 0 0.147 378 0.988 203 1.85 1.65 -38.346 074 0.027 991 0.039 586 0 0.172 959 0.983 735 1.SO 3.77 -38.447 038 0.157 325 0.222 492 0 0.806 505 0.524 687 1.50 3.75 -38.491 311 0.143 8 0.203 3 0 0.828 3 0.501 9 a All units are atomic. The coordinate system defining R and R is shown in fig. 1. Spin-coupling coefficients are numbered according to branching diagram (fig.2). As R decreases very slightly and R,simultaneously widens to 3.80a0 the spin-coupling coefficients become essentially q4= d3/2 and = 1/2. This now corresponds to the situation where o1and o3are singlet coupled and similarly o2and 04,i.e. to the formation of two C+-H bonds. We thus see in a particularly striking way the rapid change-over from reagents to intermediate as the system surmounts the potential barrier. It is worth emphasising that these large changes in the wavefunctions never give rise to any discontinuities in the total energy. This last remains smooth although it is frequently associated with a maximum as in the CH; case above. Such rapid changes in the spin coupling coefficients as the nuclear geometry varies are usually indicative of a nearby excited state which interacts strongly with the ground state.The presence of such a state is revealed as the second solution of the secular equation at the VB stage (see next section). The range of application of the spin-coupled theory is determined by the number of electrons N in wavefunction (2.1). The ‘non-orthogonality problem’ occurs in the computation of the density-matrix elements the work for which is proportional to N! This is minimised in the recurrence scheme (2.4). If there are N occupied orbitals of o symmetry and N of 71 symmetry the amount of computation is instead proportional to Nu!+N,! where N = No+N,. Core orbitals in a closed-shell con- figuration of the form {& 4 * * * 4%,,> (2.13) where Nc is the number of such orbitals play no role in this respect and any number of them may be included.Present density-matrix programs are capable of treating a problem with Nu = 8 N = 8 and N arbitrary. The actual time taken to evaluate the density matrices is negligible compared with the time required for all other steps in an iteration and the current limitations on No and N arise from the lengths of lists which it is convenient to process. Nevertheless the present limits on N and N make it possible to apply the theory directly to a wide range of molecular systems with a reasonably small number of valence electrons.* * It is worth emphasising the distinction between the number of electrons N which are treated in a non-orthogonal fashion and the total number of orbitals Nt (occupied and virtual) which might be used in a spin-coupled VB calculation.The computational labour is always proportional to N! and not to Nt!. We have for example used 35 non-orthogonal orbitals without any great difficulty in spin-coupled VB calculations on 6-electron systems. STUDIES OF MOLECULAR STATES 3. THE SPIN-COUPLED VB WAVEFUNCTIONS’ When the stabilised Newton-Raphson procedure converges all gradients are zero and eqn (2.6) becomes Its higher-energy solutions are obtained by reformulating the equation as follows where In eqn (3.3) fiP) stands fo the supermatrix of CVoperators with row and column p missing and similarly &(PI represents a diagonal matrix of all the orbital energies except cP.Eqn (3.3) is now an eigenvalue equation which can be solved in the normal way to produce an orthonormal set of orbitals 4;) and associated eigenvalues c!). Each occupied orbital gives rise to an orthonormal set (or ‘stack’) P 21 (4;) I@’)) = 6.. (3.4) the occupied orbital reappearing as one of the solutions @) say. The solutions stemming from different stacks are not orthogonal to one another and we have (4;’ 14;’)) = Api;vj 01 # v). (3* 5) Each operator Qff)is constructed from quantities involving only N-1 electrons Effectively orbital 4Pis missing and the solutions therefore describe the movement of an electron in an average field arising from just the other electrons. Consequently the virtual orbitals 4;) in general have the same semi-localised form as the occupied orbitals.Higher-energy solutions are of course more diffuse. As internuclear distances become large the virtual orbitals assume the form of excited atomic orbitals. Excited spin-coupled structures are formed by replacing 1,2,3 . . . etc. occupied orbitals by virtual orbitals The final total wavefunction is expressed as c. cc;:::I:) Y? = Wl4Z *.. 4N}+ {@$) ... $q$q (3.6) Pl ... PN i, ..,,i,v where the first term is the spin-coupled function and the succeeding terms represent the excited structures. The coefficients Co,C(::-) are determined by constructing the matrix of the Hamiltonian in this basis of spin-coupled structures and diagonalising by means of a normal VB program. From a formal point of view eqn (3.6) may be regarded as an expansion of the exact wavefunction in terms of N orthonormal sets a distinct complete set is used for each electron the particular expansion being in some sense ‘tailored’ for that electron coordinate.If the N sets coalesce into a single expansion set we regain the MO-CI or MCSCF-CI representation. This indicates that expansion (3.6) may be expected to converge reasonably quickly at least for small systems. Indeed as shown below we have found that we can achieve the same accuracy as the highest quality MCSCF-CI wavefunctions using 20-50 times fewer terms. The price to be paid of course is in the construction of the elements of the Hamiltonian matrix between the non-orthogonal D. L. COOPER J. GERRATT AND M. RAIMONDI spin-coupled structures since this requires ca.an order of magnitude longer than for a similar size matrix constructed from configurations of orthogonal orbitals. However the fact that it is now possible to attain this level of accuracy with VB-based wavefunctions has stimulated a great deal of development of our programs. Even at their present stage the times required by them per energy point are strictly comparable with that taken by large MCSCF-CI programs for closely comparable total energies. The most useful feature in our view of the compact expansion of Y given by eqn (3.6) is the physical and chemical visuality provided by it. The spin-coupled structure by itself reproduces with reasonable accuracy all the features of the ground-state potential-energy surface.Thus for example a spin-coupled wavefunction typically yields 85% of the observed binding energy and equilibrium internuclear separations are accurate to 0.01 A. This function therefore dominates expansion (3.6) at all nuclear geometries. The various excited structures provide angular and other types of correlation as an extra quantitative refinement but do not alter the qualitative picture. The same appears to be true of the low-lying excited states. The evidence we have so far shows that such states are well represented by a single excited spin-coupled structure in which one (or at most two) occupied orbitals are replaced by appropriate virtual orbitals q5t),gb$j). In other words the first few roots of lowest energy of the secular equation are dominated by just one or two structures for all configurations of the nuclei and these provide the essential physical interpretation.4. RECENT APPLICATIONSf6 4.1. ‘z+ STATES OF CH’ As indicated in section 2 CH+ is of considerable importance in interstellar chemistry. It has also been well studied in MCSCF-CI methods.l79 l* The spin-coupled wavefunction is of the form (a,a2a3a4a5o6}.Orbitals o,and a2are core orbitals and may be characterised simply as ls(C+) and ls’(C+) respectively. They change very little as R varies. Orbitals a3and o4are lone-pair orbitals. At large internuclear distances they are of the form 2s(C+) and 2s’(C+) but as R decreases they alter considerably. The same is true of the first member of the bonding pair a,. At large values of R this assumes the form of a 2p,(C+) function but undergoes considerable variation as R decreases.Its partner 06 is almost completely a ls(H) orbital at small values of R there is a significant amount of delocalisation onto C+. The wavefunction {a a 0,a4a5G6} thus furnishes a qualitatively correct description of the dissociation of CH+ into C+(ls 1s’2s2s’2p2; ,P)+H(ls). If orbitals a and a2remain coupled to a singlet there are just two spin function which give a net spin S = 0. We write these simply as 0 and 0,. Spin function 0 describes orbitals a3and a4coupled to a triplet a5and a6similarly the two triplets giving a resultant singlet. Function 0 is the perfectly paired spin function with the orbitals o3and o4coupled to a singlet and the same for a and a6.The basis set used consists of 180,20n and 66 Slater orbitals and is the same as that used by Green et except for the omission of 4ffunctions. It includes the basis functions 2p([ = 1.O) and 3p([ = 1 .O) on the H atom which are necessary to account correctly for long-range induction effects. However no diffuse 3s(C) or 3p(C) functions which would be needed to describe any Rydberg character in excited states are included. Note that at large values of R the C+ wavefunction contains a substantial contribution from the coupling ((2s 2s’) 3S;2p2}2P(spin function Q1), as well as from the expected ((242s’) lS; 2p,} 2Pstate the coefficients of the two spin functions being c 0.36 c2 % 0.93. The role of the (2s 2s’) 3S coupling is clearly to afford some STUDIES OF MOLECULAR STATES -3780 1 x 'I+ C'(*P) H -37851 h -3790 N 2 v x - 2 z -3795: -3800 1 ---- ---3805- I 1111IIIII I I I 11 1111 I11 I 1111III Ir7i111111 I111I 11111 1111II I 11 I I I 11 2 4 6 8 10 12 14 internuclear distance (a.u.) Fig.4. Potential-energy curves for CH+ XIC+ 0,spin-coupled VB 500 structures; 0, MCSCF-CI (Green et aE.17). additional radial correlation between the lone-pair electrons. At shorter interatomic distances this type of correlation becomes much less important. Spin-coupled VB calculations were carried out using a total of 26 orbitals 6 occupied 6 0 virtual and 14 n virtual orbitals. The final wavefunctions consist of 500 structures formed from 286 distinct spatial configurations of C+ symmetry.These consist of the spin-coupled referene function and (1 + 2 + 3+ 4)-fold excitations. No excitations from the (a,,a,) core were included. About half of these structures (single +double replacements) contribute to the ground state the remainder improve the description of the excited states. In fig. 4 the potential-energy curve for the XIC+state is compared with that obtained by Green et al. using an MCSCF-CI procedure. This wavefunction includes core excitations and gives the lowest energy of any calculation on CH+ in the literature. In fig. 4 this potential-energy curve has been shifted upwards so as to coincide with the spin-coupled VB result at R = 2.0~1,(the point closest to the equilibrium value of R given by the authors).The two calculations are very similar over the whole range of R.Computed spectroscopic constants for the spin-coupled VB wavefunction are shown in table 2 where they are compared with other calculations and with experiment. The excited states are shown in fig. 5. They are compared with the results of Saxon et aZ.l*by shifting the spin-coupled VB curves so as to coincide with the MCSCF-CI results at R = 15a,. The two sets of curves are remarkably similar for all values of R.The only exception occurs in the 4 lX+ state at R < 3.5~1,.The discrepancy between the two calculations in this region is due to an avoided intersection with the next highest state. This is Rydberg in character and lies 0.7 eV above the 4 lC+ asymptote yielding C(2s22p3s;'P)+ H+ on dissociation.Since no diffuse 3s or 3p basis functions were included this state is absent in our calculations and the divergence with the MCSCF-CI result occurs just where it would intersect the spin-coupled VB state. D. L. COOPER J. GERRATT AND M. RAIMONDI Table 2. Properties of CH+(XIZ+) 3.63 spin-coupled -37.950 56 1.141 276 1 60 4.14 spin-coupled VB -38.024 26 1.135 2845 69 MCSCF-CI” -38.060 64 1.130 2847 63 4.1 1 4.14 MCSCF-CI” -38.022 33 1.128 2860 59 experimentalz9 -1.131 2858 59.3 4.26 -371 2 -374 0 v ;h C’(‘D ) H 2 -376 4 C(’S) HI + -3 3 -377; 3 3 ~ -378 7 I. I I I I I, I I I I I, 2 4 6 8 10 12 14 internuclear distance (a.u.) Fig.5. Potential-energy curves for excited lZ+ states of CH+ 0,spin-coupled VB 500 structures; 0, MCSCF-CI (Saxon et~11.l~). 4.2. lZ+ STATES OF LiHe+l9 The spin-coupled wavefunction for this system can be written as {‘T~ o3a,}. ‘T~ Orbitals crl and o2 stem from the Li+ ion and can be characterised as ls(Li+) and ls’(Li+) respectively while orbitals ‘T and ‘T can be regarded as ls(He) and ls’(He). There are two spin functions corresponding to singlet states these being the same as those used above for CH+(O and OJ. Consequently the ground state of this ion corresponds simply to the interaction of the two closed-shell species Li+(ls 1s’; ‘S) and He( Is 1s’; lS). However as soon as a single electron is excited out of either atom a whole series of closely spaced states arises.These differ widely in character from one another. Their corresponding potential-energy curves show several avoided crossings and the associated non-adiabatic radial couplings give rise to charge exchange in low-energy collisions. The description of all of these states by spin-coupled VB theory furnishes a test of the quality of the virtual orbitals and affords us valuable experience with a small system in the development of procedures for the selection of the necessary structures and orbitals. STUDIES OF MOLECULAR STATES Table 3. LiHe+‘C + states comparison of experimental asymptotes with calculated energies at 30 bohr from a 23-structure spin-coupled VB calculation as described in the text energy/eV state asymptote experimental2* calculated (1) 1C+ (2) 1E+ (3) 1c+ (4) lC+ (5) 1c+ (6) IC+ Li+(lsls’)He(lsls’) Li(lsls’2s) He+(ls) Li+(1 s 1s’) He( 1 s2s) Li( 1s1s’2p) He+( 1s) Li+(1 s 1 s’) He( ls2p) Li(lsls’3s) He+(ls) -19.190 0 1.425 1.848 2.028 3.373 -18.752 0 1.454 1.843 2.049 3.676 The basis set employed is a large ‘universal even-tempered ’ set of Slater functions,20 comprising 420 and 44n orbitals.Using this converged spin-coupled wavefunctions were obtained at 24 internuclear distances between la and 30a,. Initial spin-coupled VB calculations were carried out usingjust 23 structures formed from 16 orbitals and the asymptotic energies are shown in table 3. Note that since a theorem analogous to that of Brillouin2’ applies to the spin-coupled wavefunctions few of these structures interact with one another and each state is described essentially by just one or two structures.As can be seen the calculated asymptotic splittings are in good agreement with experiment. It is worth remarking that by using such a large basis set the virtual orbitals are more flexible and consequently fewer of them in fewer structures seem to be required. The final calculations include 14 virtual orbitals of CT symmetry and 16 of z. All single and double replacements from o3and o4(Is 1s’of He) only were included giving rise to 192 spatial configurations. The orbitals o1and 0,(Is Is’of Li+) were regarded as core and not excited. However several virtual orbitals stemming from the o1and o2stacks were included in the calculations.The lowest seven lC+ states are shown in fig. 6. There are multiple avoided crossings of states which correspond to Li++He( Is nl) and Li( Isls’ nZ)+ He+(1s). The asymptotic splitting between the first two states is now 19.03eV. The lowest state possesses a shallow minimum 843 cm-l deep with equilibrium internuclear distance of 1.930A owing to inductive and van der Waals effects and which supports six bound levels. The first excited state shows a much deeper minimum of 4979 cm-l with Re = 3.587 A and gives rise to ca. 28 bound states. The use of such a large number of non-orthogonal orbitals to expand essentially just two electronic coordinates brings with it the hazard of near linear dependence. In practice this arises from large overlaps (> 0.998) between virtual orbitals at certain internuclear separations.We have found that the most effective remedy is simply to orthogonalise the offending orbitals amongst themselves using the Schmidt procedure.22 If two such orbitals are denoted by u and v then as long as all structures of the type {... uu ...I,{... u2...> {... u2 ...} are included the total wavefunction is invariant to such a transformation. As a result the final wavefunctions and potential-energy curves are free from any unwanted effects arising from incipient linear dependence. However there remain some small variations in the lowest root (of the order of a.u.) in the region of R = 8.5a0.We ascribe this to basis set superposition error which we have not attempted to correct.D. L. COOPER J. GERRATT AND M. RAIMONDI -90j cJ -92j 41 v x 2 -94 E -96 -981 -100 -102 €I 5 10 15 20 25 30 internuclear distance (a.u.) Fig. 6. Potential-energy curves for excited lC + states 192 structures. of LiHe+ spin-coupled VB 4.3. POTENTIAL-ENERGY SURFACES FOR THE B+ + H SYSTEM PRELIMINARY RESULTS The reaction B++H + BH++H has been studied by Friedrich and Herman23 and by Ottinger and Reichm~th.~~ The incident beam consisted of a mixture of B+ ions in the lS and in the metastable 3P states. Emission from product ions was observed and assigned to the transitions BH+(A ,lT+ X2C+)and BH+(B T++ X2C+).The existence of the bound B zC+ state was previously unknown. It was confirmed by the calculations of Klein et uI.,,~ and correlates well with our own findings for the corresponding state of BeH.g Preliminary ab initio calculations on the BH system have been carried out by Hirst26 using the MRD-CI method and extensive DIM calculations of both singlet and triplet surfaces have been reported by Schneider et uI.,~ The aim of the present spin-coupled VB work is to study the potential-energy surfaces corresponding to the processes /BH+(A2n; B2X+)+H B+(1S13P)>B(2P) + H;(,X;) LBH(x~x+)+H+.The calculations were initiated with all three atoms well separated in order to select orbitals and structures which would reproduce reasonably well the observed energy differences between all the different possible asymptotic states. A large Gaussian set is used consisting of B( 1 1,6,2/9,5,2) and H(6,2 1 /4,2 1) on each H atom.This set is taken from ref. (25) and (30) and is well suited for study of both BH+ and BH. It 6 FAR STUDIES OF MOLECULAR STATES Table 4. Asymptotic splittings for the B++H system separation/eV total energy state (a.u.) computed experimental ~ B+(lS)+H(T) +H(?S) B+(3P)+H(,S) +H(2S) B(2s22p;2P)+H++H(,S) B+('P) +H(,S) +H(2S) -25.308 46 -25.135 76 -25.105 90 -25.104 96 0.0 4.71 5.52 9.44 0.0 4.63 5.31 9.09 also includes diffuse functions on the B atoms which are needed to represent the B(2s23s;2S)Rydberg state. A set of 342 structures was chosen consisting of all single and double excitations from the occupied orbitals and also single and double excitations from certain singly excited configurations which are selected to represent the different excited and charge-transfer states.This set of structures yields values for the asymptotic splittings shown in table 4. The experimental separations are thus reasonably well reproduced. Experience by ourselves and othersls shows that a further increase in the number of structures serves only to lower each state by a uniform amount which is almost independent of nuclear geometry i.e.the remaining deficiencies in the selected set of structures are almost completely atomic in origin and play little or no role in the behaviour of the molecular potential-energy surfaces. Further results on the potential-energy surfaces of this system will be presented at the Symposium.B. H. Lmgsfield A. D. McLean M. Yoshimine and B. Liu J. Chem. Phys. 1983,79 189 G. A. Gallup and J. R. Collins to be published. J. Gerratt in Theoretical Chemistry (Specialist Periodical Report The Chemical Society London 1974) vol. 1. C. A. Coulson and I. Fischer Philos. Mag. 1949 40,386. J. Gerratt Adv. Atom. Mol. Phys. 1971 7 141. N. C. Pyper and J. Gerratt Proc. R. Soc. London Ser. A 1977 355 407. ' M. Kotani A. Amemiya E. Ishiguro and T. Kimura Tables of Molecular Integrals (Maruzeu Tokyo 1963). J. C. Manley and J. Gerratt Comput. Phys. Commun. 1984 31 75. J. Gerratt and M. Raimondi Proc. R. Soc. London Ser. A 1980,371 525. lo S. G. Walters Ph.D. Thesis (University of Bristol 1984). l1 A. Dalgarno and J. H. Black Rep. Prog. Phys.1976 39 573; M. Elitzur and W. D. Watson Astrophys. J. 1980 236 172. l2 D. H. Liskow C. F. Bender and H. F. Schaefer J. Chem. Phys. 1974 61 2507. l3 P. K. Pearson and E. Roueff J. Chem. Phys. 1976 64 1240. l4 S. Sakai S. Kato and K. Morokuma J. Chem. Phys. 1981,75 5398. l5 J. Gerratt S. G. Walters and R. Williams to be published. l6 J. Gerratt J. C. Manley and M. Raimondi J. Chem. Phys. in press. l7 S. Green P. S. Bagus B. Liu A. D. McLean and M. Yoshimine Phys. Rev. A 1972 9 1614. R. P. Saxon K. Kirby and B. Liu J. Chem. Phys. 1980,73 1873. l9 D. L. Cooper J. Gerratt and M. Raimondi to be published. 2o D. L. Cooper and S. Wilson J,Chem. Phys. 1983 78 2456. 21 L. Brillouin Act. Sci. Ind. 1934 159. G.A. Gallup personal communication ;R. 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