Faraday Symp. Chem. SOC.,1984 19 97-107 Current Status of the Multiconfiguration-Configuration Interaction (MC-CI) Method as Applied to Molecules Containing Transi tion-metal At oms BY PERE. M. SIEGBAHN Institute of Theoretical Physics University of Stockholm S-11346 Stockholm Sweden Received 14th August 1984 The current status of the MC-CI method is reviewed. Calculations on transition metals show that the MCSCF step is often the most problematic part of these types of calculations. The reason is that the most desirable MCSCF calculation can easily be too large for presently available methods. The limitation lies in the number of configurations which can be handled. A new direct CASSCF CI method is suggested which is capable of efficiently handling a larger number of configurations.In the multi-reference CI step the limitation is often in the size of the reference space rather than the total number of configurations. Methods are therefore needed in which the internal space is more efficiently handled particularly with respect to the number of formulae which need to be stored. The new CASSCF CI method can also be useful in this context. 1. INTRODUCTION In the multiconfigurationsonfiguration interaction (MC-CI) method the zeroth- order subspace is selected by an initial MCSCF calculation. The most important configurations in the MCSCF calculation are then selected as reference states for a final singles and doubles CI calculation. In this paper the current status of the MC-CI method will be described.Examples of the models used and problems encountered in applications to molecules containing atoms of the second and third rows of the periodic table will be given. Based on these calculations possible directions of future developments of the method are discussed. It is of interest in this context to review the MCSCF and CI methods. This history which is given in the first section of the paper will of necessity be very brief and incomplete. Only some of the most important references will be mentioned. Since the ideas behind the MCSCF and CI methods are very simple they were suggested soon after the development of quantum mechanics in the early 1930s. Owing to the lack of sufficiently powerful computers the applications of these methods to molecules did not appear until the end of the 1960s.The first applications were naturally on diatomic molecules. The early MCSCF calculations had a CI expansion of only a few terms and the method used was essentially an extension of the Roothaan open-shell SCF method with Lagrange multipliers.l9 The restriction to a few configurations was the consequence of an over-complicated method rather than a technical limitation of the computers available. In the early 1970s Levy3 showed that the use of an exponential form of the unitary transformation in connection with the Newton-Raphson formalism simplified the MCSCF formalism drastically. The general formulation of the MCSCF method is today essentially the same as outlined in ref. (3). One line of the modern development of the CI method essentially followed the 97 FAR CURRENT STATUS OF THE MC-CI METHOD general strategy outlined by Boys in the early 1950~.~ This method is known today as the conventional CI method and is almost unchanged.In this method the formulae for the Hamiltonian matrix elements are first constructed and stored. With these formulae the Hamiltonian matrix is set up and diagonalized by iterative techniques. The method is characterized by a high flexibility and a rather low upper limit for the allowed number of configurations of the order of lo4 terms. The most efficient variant of this method is probably the MRD-CI program by B~enker,~ where perturbation selection and perturbation extrapolation are two key features. The other line of evolution of CI methods started when the development of efficient general molecular Hartree-Fock programs was essentially completed in the early 1970s.A natural zeroth-order starting point was then a single Hartree-Fock determinant. To account for the major part of the correlation energy a wavefunction composed of single and double excitations from the Hartree-Fock determinant a HF-SD wavefunction was considered. That single and double excitations are the most important excitations for the dynamical correlation of the motion of the electrons follows directly from the two-particle nature of the electron-repulsion operator. Two efficient methods emerged very early in the solution of the CI problem for an HF-SD wavefunction. The PNO-CI method of Meyer6 used the simple structure of the two-electron problem to reduce the CI expansion to only a few hundred non-orthogonal terms.In the direct CI method the HF-SD CI problem was solved without constructing the Hamiltonian matrix explicitly.' The PNO-CI method of Meyer had an additional feature which has turned out to be very important for achieving a high accuracy. The CEPA correction for unlinked clusters was introduced based on the CP-MET theory of Cizeks and the work of Sinan~glu.~ With the CEPA correction a link was introduced between the variational CI approaches and the MBPT approaches. During the mid 1970s the cluster methods were developed towards the full solution of the CP-MET equations which was first done by Taylor et aZ.l0 A still simpler approximation than the CEPA correction is Davidson's often used correction.11 The fact that cluster corrections usually improve the results is now generally accepted and most CI calculations made today include such corrections.The question of size consistency is therefore no longer a dividing line between CI and MBPT methods. Since the early 1970sand the first efficient HF-SD methods the further development of methods for the electron correlation problem has gone in two different directions. First CI methods have been developed with a more general MCSCF zeroth-order starting point. Secondly MBPT methods have been developed which go beyond third-order perturbation theory. The essential equivalence between the first iterations of CI and MBPT up to third order was realized early in the 1970s [see e.g.ref. (12)]. In the direct CI program second-and third-order Moller-Plesset perturbation-theory results were routinely generated in all CI calculations. Note that these types of calculations did not become popular among ordinary chemists until recently almost ten years later when essentially the same algorithms were implemented in the GAUSSIAN 80 package.13 The first HF-SD programs already contained most of the features available in MBPT programs today. Apart from the very recent development of multireference coupled-cluster methods the main new development of the MBPT method has been the inclusion of the full fourth-order contribution. In going beyond the HF-SD approximation the emphasis has been in CI language on the inclusion of the major effects of triple and quadruple excitations.This has been combined with a requirement of size consistency which is however also required approximately in most CI calculations today. The major advantage of the MBPT approach in going beyond the P. E. M. SIEGBAHN 99 HF-SD approximation is that it is a well defined procedure and convenient to use. It is easy to predict that this is the procedure which will be of most use to chemists in high-accuracy applications. These calculations do not necessarily require expert quan tum chemists. The development of CI methods during the 1970s focused on an improvement of the zeroth-order wavefunction. The most important configurations usually described in terms of MCSCF orbitals define the reference space for single and double excitations.Most of the technical problems involved in the application of this procedure the MC-CI method were solved with the generalization of the direct CI method.14 The major disadvantage with this approach is that MCSCF is not always an easily defined procedure and can be very difficult to apply correctly by the user. This is particularly true when individual configurations should be selected. When orbital spaces should be selected as in the CASSCF method,15 the method can be made into a routine procedure if large enough CI expansions can be handled. When this is not the case CASSCF calculations still require expert quantum chemists. Note that if the MCSCF step is left out the multi-reference CI method can be made into a routine meth~d.~ One might well ask if it is necessary or desirable to use the MCSCF method.This question and others connected with applications of the MCSCF method are addressed in the next section of this paper. In section 2 it is argued that it is desirable in many cases to be able to make very large MCSCF calculations. The practical problem concerning this is found in the CI section where no efficient method exists today capable of making large general CI expansions. The only type of very long CI expansion which can be treated is the multi-reference SD expansion. However this type of wavefunction goes against the general idea of the MC-CI method in which all configurations in the zeroth-order space should be treated equally.The bottleneck to large general CI expansions is the large number of formulae needed on peripheral storage. This is the same problem that -prevented long HF-SD expansions before the direct CI method and should be solved by similar techniques. A method of treating long CASSCF expansions is discussed in section 3. Some applications of the MC-CI method to molecules containing transition metals are discussed in section 4. The emphasis is on the problems encountered in the applications and possible future developments suggested by these problems. It is predicted that the future development of the MC-CI method will be concerned with reducing the storage requirements for formulae still further. This requires a restructuring of the internal-space treatment.Some of the ideas used in the new method for large CASSCF expansions described in section 3 may be useful in this context. 2. COMMENTS ON MCSCF APPLICATIONS In this section some of the advantages and disadvantages of the MCSCF method as it is used today are discussed. Results from calculations on molecules containing second- and third-row atoms will be used to illustrate the points being made. The first question is how MCSCF should be used. This is a matter of choice. One way the MCSCF method has been successfully used is by trying to select only a few important configurations as in the OVC method of Wahl and Das for example.2 This procedure requires the user to have considerable experience and was developed at a time when only a few configurations could be handled technically.Today it is easily possible to select larger classes of configurations and this should almost always be an advantage. This type of method will therefore probably disappear just as methods which select only certain diagrams in MBPT have disappeared. The other extreme use 4-2 CURRENT STATUS OF THE MC-CI METHOD of MCSCF is the selection of all configurations in a certain orbital space as in the original CASSCF method.15 In this procedure the user should select orbitals rather than configurations which is of course much easier. Two well defined procedures can be set up for the orbital choice. The first procedure is the full-valence MCSCF where all configurations in the valence space are selected. The second preferable procedure is to select one weakly occupied orbital for each strongly occupied orbital.For H,O this leads to the addition of two orbitals outside the valence space and provides much improved results compared with the full-valence MCSCF procedure.16 The main problem with these simple well defined procedures is that the number of configurations increases rapidly with the number of valence electrons and is often unmanageable. One way to reduce the number of configurations without destroying the simple structure of the approach is to fix the number of electrons in each symmetry. In a 12-electron valence CASSCF on Cr this procedure leads to a reduction in the number of configurations by an order of magnitude without significantly affecting the accuracy. Even with this reduction in the number of configurations these well defined MCSCF approaches still lead to unmanageable CI expansions in many cases.Further reductions in the orbital space or configuration space will often require much experience of the user and is why the future of the MCSCF method as a standard tool for chemists depends very much on the possibility of being able to treat longer CI expansions. A method of treating longer expansions is described in section 3. The second question is when MCSCF should be used. The first area of application is trivially the cases where the single-configuration SCF method gives bad results. Such situations are rare for ground-state molecules containing only first-row atoms. For molecules with second-row atoms SCF results are often poor and with atoms of the third row it is more common for SCF to give bad results than reasonable ones.” For a small stable molecule like NiCO the binding energy is negative by 59 kcal mol-1 when the correct value should be 30 kcal mol-l.The wrong ground state is also predicted. For Fe(CO) and Fe(C,H,) the iron-ligand bond distance is incorrect by ca. 0.5 a.u. even though the bond (particularly for ferrocene) is very strong. These large errors are essentially corrected with a rather small CASSCF calculation where the active space is selected with the most preferable method described above i.e. one weakly occupied orbital for each strongly occupied orbital. The iron d orbitals are in this case either weakly or strongly occupied which leads to 10 electrons in 10 orbitals.It was also our experience in the case of Fe(CO) that it was nearly impossible to define a multi-reference CI based on the SCF orbitals and obtain a reasonable bond distance.18 For the seemingly simple diatomic molecules ClF and CCl in their ground states the SCF approximation also gives rather poor results. In these cases the bond distances are nearly 0.1 a.u. too short at the SCF level. A large MCSCF including the chlorine 3dshell was needed to obtain high-accuracy resultslg (see the discussion below). Again to extend this treatment to larger systems methods of handling large CI expansions in the MCSCF method are needed. The calculations described in the preceding paragraph have been examples of cases where an initial MCSCF is needed even when a multi-reference CI calculation is performed afterwards.There are other situations where an MCSCF calculation is needed and where a subsequent MC-CI calculation is not necessary. For molecular properties depending critically on the charge distribution such as multipole moments and transition moments a large MCSCF is sometimes required and is also sufficient.20 A CASSCF-type wavefunction also has invariance properties which greatly simplify the evaluation of the non-orthogonal transition moment. Another area where MCSCF will be very important in the future mainly for technical reasons is in the optimization of geometries of stable molecules and of transition states for reactions. Since the P. E. M. SIEGBAHN explicit derivatives with respect to geometrical parameters are much easier to calculate for wavefunctions where all the parameters of the wavefunction are variationally optimized the MCSCF method has a big advantage compared with CI methods.It is also our experience that a well defined MCSCF of the type discussed above will give accurate geometries. This is true even for transition-metal compounds with Cr as a notable exception. If saddle points for reactions involving transition metals are to be located with derivative methods very large CI expansions will again be required since the wavefunction commonly has quite a different form on one side of the barrier than on the other. Wavefunctions which go smoothly over the barrier are required to give meaningful derivatives.The final and in our opinion also the most important reason to carry out MCSCF calculations concerns the understanding of a given problem. Whenever an understanding is the issue the viewpoints will by necessity be very personal. Let us illustrate the arguments with the following calculations on CC1. The SCF and MC-CI calculations have been made.19 The other results from an HF-SD calculation are hypothetical and extrapolated from similar two-configuration reference CI calculations on CC1. The bond distance in CCl is too short at the SCF level by ca. 0.1 a.u. An HF-SD calculation using the SCF orbitals would as usual make the bond distance longer but probably still too short. A cluster correction of for example the CEPA type would then normally lead to an additional increase in the bond distance and it would not be surprising if the final result were close to the experimental value.Except possibly for the coefficient of the bond-dissociation configuration all coefficients in the wavefunction will be small of the order of 0.05 or less. Coefficients of this size appear for all molecules. The correct bond distance would be a valuable result and would show the capability of this simple well defined method. Let us now move to MCSCF calculations which have actually been performed. First the simple two- configuration MCSCF calculation which allows for proper dissociation gives a bond distance equally far from the experimental bond distance as the SCF result 0.1 a.u. but this time too long. This is a common effect.A full-valence CASSCF calculation including only the s and p shells gives the same poor result. More surprising is the fact that a multi-reference MC-CI calculation using the most important reference states does not significantly change this result. However including also the 3d shell of chlorine in the CASCF calculation has a dramatic effect on the bond distance and on the potential curve. Suddenly all spectroscopic constants come out in good agreement with experiment. Excellent agreement with experiment is finally obtained after an MC-CI calculation based on these latter CASSCF orbitals. Note that the 3d shell of chlorine is also very important to obtain an accurate ionization potential and electron affinity of the chlorine atom. Essentially the same sequence of results as for CCl was also obtained for ClF.By far the most important result of these MC-CI calculations in our opinion is the qualitative and general understanding of the effect of the chlorine 3d shell. This effect is not seen at all in an HF-SD +cluster-correction calculation which may otherwise show the same type of accuracy for all measurable properties. A few comments on the technical aspects of the MCSCF method may also be of interest. Powerful techniques exist today for converging MCSCF calculations [see for example ref. (21) and references therein]. Most of these quadratically convergent procedures have been implemented in our program. It may therefore be surprising that in practically all the applications we perform the simple first-order approximate super-CI method15 is the method which is used.The reason is that this method is in general much faster than the more elaborate methods particularly for large basis sets. A calculation on NO is illustrative. The first-order method required < 5 min per CURRENT STATUS OF THE MC-CI METHOD iteration and converged in 7 iterations. A second-order method took > 1 h for every iteration and was therefore not taken to convergence. Practically all of this time is taken up by the integral transformation. In our applications we have therefore only used second-order methods when the first-order method does not converge at all or for cases with few electrons. For cases where a high degree of convergence in the energy is required as for example when certain one-electron properties are calculated second-order methods may also be needed.Summarizing this section the MCSCF method is mainly useful for the insight it gives into the chemistry of problems and also for certain technical advantages which are achieved from invariance properties (transition moments) and the variationally optimized orbital parameters (geometry optimizations). It is further almost necessary to use an MCSCF starting point for multi-reference CI calculations whenever the SCF approximation gives bad results as is often the case for transition-metal compounds. In the systematic improvement of a multi-reference wavefunction where the SCF approximation still works reasonably well as for molecules containing first-row atoms MCSCF can also be of importance.However this is more debatable and depends on a given case. A straightforward and systematic application of the multi-reference CI methods based on SCF orbitals can be more effi~ient.~ Finally a systematic and well defined MCSCF approach which can generally be used for a large variety of problems will require the possibility of treating long CI expansions. A development in this direction is described in the next section. 3. A NEW CASSCF CI METHOD The direct CI method described in this section should be used as part of an MCSCF program and the goal is to perform large CI expansions. For this to be possible a minimum number of formulae must be stored. A secondary goal is that the algorithms should vectorize well.The method must also be able to treat certain reductions of the CASSCF expansions such as a limitation on the number of electrons in each spatial symmetry. The description of the method will be brief and the reader is referred to ref. (22) for further details. In the CI problem the Hamiltonian matrix is diagonalized either by an iterative method or by perturbation theory. In both cases the main computational step in each iteration is the construction of the vector o as where p,v are labels for the chosen configuration basis and c is the wavefunction from the previous iteration. The main working equation in the direct CI method can then be written as where and are the direct CI coupling coefficients. These expressions follow from the usual form of the Hamiltonian operator in its second quantized form where the Epq are the generators of the unitary group.From the expressions for thesecoupling coefficients it is obvious that the two-electron coefficients can be written as products of one-electron coefficients. The exact P. E. M. SIEGBAHN expression is obtained by inserting the projection operator over the complete configuration space the resolution of the indentity In the present method we use the fact that in the product in eqn (2) the same intermediate index K appears in both one-electron coupling coefficients. Only the one-electron coefficients are calculated and stored and these coefficients are ordered after K. This means that when the two-electron coefficients are constructed in the update formula (1) they can in principle be formed directly from multiplications within the same group of one-electron coefficients which are ordered sequentially.These multiplications are however only done indirectly (see below). When eqn (2) is inserted into eqn (1) we obtain for the term involving the product of one-electron coupling coefficients Aop = f C X A% AF:(pg I rs) c,. v,pqrs K This term can then be written after reorganization as which has been written to emphasize the matrix structure with Drs,K= C AFXc,,. 1’ Eqn (3) in matrix notation is written as Aop = iTr(Ap.1.D). The corresponding contribution AP to the second-order density matrix P which is needed in the CASSCF method is written as AP = +D*D’.The three matrices in eqn (3) have different characters. The symmetry-blocked integral matrix I is completely dense i.e. there are normally no zero elements. The matrix AP on the other hand is very sparse. The matrix D is in a normal CASSCF calculation rather dense; ca. 50% of the elements are non-zero. The trace of a product of three matrices requires one matrix multiplication and a scalar product. The matrix product can be performed between any two of the three matrices. Since the matrices D and I are independent of the index p it is clearly an advantage to perform the product between these two matrices. Some properties of the D matrix are worth noting. First it is straightforward to store the coupling coefficients required to form one matrix element of D sequentially on the formula tape which makes the formation of the D matrix easy.Secondly when there are many different spin-coupling possibilities there will be many CI coefficients c which will go into the same matrix element of D. This summation over spin couplings will therefore drastically reduce the number of necessary multiplications required to form o. This is an advantage which is not present in conventional CI techniques where each matrix element is multiplied by only one CI coefficient during the iterations. CURRENT STATUS OF THE MC-CI METHOD A detailed timing analysis of the present method as formulated in eqn (3) and a comparison with the conventional approach as formulated in eqn (1) is quite difficult. As mentioned in the introduction the goal with the present method was to reduce the elapsed time and more important the storage requirements.Such a reduction is sometimes worth even a large increase in C.P.U. time. (Compare this for example with the philosophy in the integral program DISCO where the integrals are recomputed in each SCF iterati~n.,~) It may seem as if the conventional approach must have an advantage in terms of C.P.U. time compared with the present approach since the summation over K in eqn (2) is performed prior to the CI step. This sum which is over the different spin couplings can in normal CASSCF calculations often be quite long owing to the large number of configurations with many open shells. There are two arguments against this simplified way of reasoning.First the C.P.U. time spent in reading and unpacking a large number of two-electron coupling coefficients is certainly not negligible. Secondly in the present method a presummation over spin couplings before the multiplication with the integrals is in any case made when the matrix D is formed. Conclusive timing comparisons between the two methods have not been made yet since optimal versions of the two methods are not available on the same type of computer. Preliminary experience shows however that on a vector processor the present method is usually much faster. On scalar machines the present method is also faster in most cases. Exceptions are cases where the number of configurations is substantially reduced owing to symmetry and excitation levels between symmetries in which case the methods are about equally fast.The largest case tried so far was a 30000-configuration calculation on Ni(C2H4)2 where the two-electron part required 10 s per iteration on a Cray-1 computer. Half of this time went into reading the one-electron coupling coefficients. A goal for further improvements of the method is therefore to avoid storing these coefficients. Algorithms for constructing these coefficients directly in the required order is under development. With the use of prototype formulae this turns out to be straightforward. 4. MC-CI CALCULATIONS ON TRANSITION-METAL SYSTEMS To perform calculations of qualitative accuracy on molecules containing transition metals is difficult with present techniques and will probably remain so for quite some time.To illustrate some of the problems appearing in these types of calculations two examples on rather small systems will be discussed. One system is NiCO where the ground state has only recently been determined. For this system it was possible following a reasonably well defined approach to design calculations which were small but which still yielded results of qualitative accuracy. The other system is FeO, for which present techniques are used to their limitations. Before we go on to discuss these examples the basis-set problems should be mentioned. For molecules containing only a few atoms of the first row it is today possible to saturate the basis set. This usually means much larger basis sets than are usually used however.Even if one atom in the molecule is from the second row basis-set saturation can be achieved. For molecules containing atoms of the third row this is no longer possible and is why calculations on molecules containing transition metals are only aimed at qualitative accuracy. Errors of 0.03 a.u. in the bond distances and 25% errors in the binding energies are what can normally be expected. Linear NiCO which is the most stable form of the molecule has often been used as a model for a complex with a typical transition-metal-ligand bond. It can also be used to simulate successfully bonding of molecules in positions on top of transition- metal surfaces. A one-configuration SCF treatment leads (as mentioned in section 2) P. E. M. SIEGBAHN to very poor descriptions for some of the states including the ground lC+ state which will be considered here.The binding energy of this state is negative by 59 kcal mol-1 when it should be positive by at least 30 kcal mol-l. The 3A state is better described at the SCF level and is why this state has earlier been predicted to be the ground state. The occupied orbitals of the carbonyl ligand are nearly unchanged compared with the free CO molecule in NiCO. This fact greatly simplifies the MC-CI treatment of this system. A reasonable description at the CASSCF level will be obtained without any strongly occupied ligand orbitals in the active space. A straightforward appli- cation of the most preferrable method of selecting orbitals as described in section 2 with one weakly occupied orbital for every strongly occupied orbital leads to 10 electrons in 10 orbitals.There is no problem to perform this CASSCF calculation particularly when spatial symmetry is used. As it turns out the two da orbitals stay practically doubly occupied so the CASSCF calculation could in fact have been per- formed with only 6 active electrons in 6 active orbitals. This is the minimum active space which gives a reasonable description for this state. Fixing the number of electrons in each symmetry to the Hartree-Fock occupation also works well in this case which leads to a compact description of the wavefunction at the CASSCF level. In this wavefunction there are 7 configurations (C,,) with coefficients > 0.10. No other coefficient is as large as 0.05.A multi-reference CI calculation with 7 reference states and a reasonably large basis set leads to (1-2) x lo5 configurations and the calculation requires ca. 2 h on a VAX 780 mini-computer using the contracted CI method.24 This is consequently a case of an unproblematic but by usual standards still rather large calculation. A very common complication is otherwise that those d orbitals which are not correlated in the CASSCF calculation will become mixed with ligand orbitals. These orbitals have to be cleaned from ligand contamination by some localization procedure. In our group we have simply used stepwise two-by-two rotations between the orbitals which are mixed to maximize the d contribution into as few orbitals as possible. This procedure has worked very well and constitutes a standard finishing step of each CASSCF calculation.Without this step almost random numbers will often be generated in the following CI calculation. A CI calculation correlating the d orbitals as they come out of a standard SCF calculation is consequently often meaningless for transition-metal compounds. A calculation on FeO is much more complicated than one on NiCO. In FeO the conformer with the ligand in a side-on orientation is more stable than that in a linear end-on orientation. In the side-on orientation the catalytic properties of the transition metal are more noticeable and the 0 ligand bond has started to break. This means that bonding-to-antibonding excitations on 0 cannot be disregarded. Another even more severe difficulty in the calculation on FeO as compared with NiCO is that the mixing between atomic states is in general more complicated on iron than on nickel.For NiCO in the ground lC+ state the only atomic state present on the Ni atom is the lD state with occupation d9s. This is a state which is favourable for bonding and is also a rather low-lying state on the atom. The corresponding 3F state on the iron atom with occupation d7sis far above the lowest 5D state which has an occupation of d6s2.Large mixings between d7and d6 states will therefore occur and it is not even clear what the spin state will be for FeO,. NiO is therefore simpler to treat than FeO,. Ideally a CASSCF calculation on FeO would include the 3d and 4selectrons on iron and the valence electrons (except 2s) on oxygen as active electrons.A full-valence CASSCF calculation is in this case at the limit of what can presently be handled. With the recipe of one weakly occupied orbital for each strongly occupied this leads to a CASSCF CI with 16 electrons in 16 orbitals which is an unmanageably long list of configurations. Whenever this situation occurs i.e. that the desirable calculation is CURRENT STATUS OF THE MC-CI METHOD outside the limitations of the program the user has to make a large series of test calculations before a reasonable calculation can be set up. The larger the CASSCF that can be handled the easier it is for the user which is why a method such as that described in the preceding section is particularly useful for transition-metal systems.Once a large CASSCF calculation has been done it is usually easy to design smaller calculations which also give reasonable results. The CASSCF calculations on transition-metal systems must always be followed by MC-CI calculations with as many reference states as possible. This is not only because dynamical correlation effects are generally important for these systems but also because such a calculation constitutes a test of the accuracy of the preceding CASSCF calculation. If a configuration outside the reference space appears with a large coefficient the calculation should preferrably by redone. It is our experience that if this configuration has an orbital occupied which is outside the CASSCF orbital space it is not enough to redo the CI step with this additional reference state.The CASSCF step also has to be redone including this extra orbital. Even when the coefficients for configurations including this orbital are as small as 0.05 in the CI calculation the inclusion of this extra orbital in the CASSCF calculation can completely change the bonding situation. An example is from a calculation on the 3B state of FeO, where the inclusion of a certain weakly occupied orbital in the first symmetry was critical. When this orbital was not active in the CASSCF step iron had a 3d occupation of 7.0 and 4s occupation of 0.2. The MC-CI calculation did not change these occupations significantly and no configurations with this orbital occupied appeared with coefficients as large as 0.05.Redoing the CASSCF calculation with this orbital active gave a 3d occupation of 6.4 and a 4s occupation of 0.8. The added active orbital had an occupation of 0.3. This is consequently a case where the MC-CI calculation did not give a clear indication that something was wrong. This can happen but is not normally the case. The only unusual feature of the MC-CI calculation was that the sum of the occupations for the orbitals in the first symmetry outside the reference space was as high as 0.05. The reason that there was no large coefficient in the MC-CI calculation is of course that the important orbital in the larger CASSCF calculation was spread out over very many orbitals in the smaller CASSCF calculation. To some extent this will always happen and is why a CI or MBPT calculation is so much harder to interpret and understand than an MCSCF calculation.The risk is obvious that all the chemistry the CI calculation will provide is the calculated number. A technical problem is also of general interest in the MC-CI calculations on FeO,. The 3B state which is one of the candidates for the ground state has 13 con- figurations with coefficients > 0.09 and a very large number of configurations with coefficients > 0.05. This means that a proper MC-CI calculation will have a very large number of configurations. This is however not the main problem in performing this calculation. What is even more problematic is that the reference states have varying occupations in at least 8 orbitals. This means that the evaluation of the formulae for the internal space will be time-consuming and even worse will require a large storage space.This calculation is in fact on the limit of what can be done today with the present direct CI technique. This is consequently the part where further method development of the multi-reference CI method is most urgently needed. Ideally the formulae for the internal space should be calculated during the CI iterations and never be stored. Work in this direction will most certainly be done the next few years. The method described in section 3 can also be useful in this context. It can be used without modifications for the formulae for the all-internal integrals. What remains is to develop the method further for the formulae for the integrals with one external index which presently requires most of the time and storage space.P. E. M. SIEGBAHN 107 Summarizing this section on MC-CI calculations on transition-metal compounds it is clear that many systems of chemical interest can be treated following simple principles. NiCO is an example of such a molecule. For partly dissociating systems such as FeO the situation is more difficult. What is required is MCSCF methods capable of handling a large number of active orbitals and MC-CI methods capable of treating large reference spaces. Both of these bottlenecks have to be removed by calculating the necessary formulae as they are needed without storing them. CONCLUSIONS In this paper the present status of the MC-CI method has been described.Examples are given particularly for molecules containing transi tion-metal atoms to illustrate the capability and the problems with the method. It is shown that the MCSCF calculation is the step which is most problematic. Well defined orbital selection schemes in the CASSCF method often require large orbital spaces. Methods for treating large CI expansions in the CASSCF method are therefore urgently needed. One such method is under development and was described in section 3. For the multi-reference CI step the main problem is usually not the total length of the CI expansion. Method development is instead required in order to allow larger reference spaces. This means an improved treatment of the formulae for the internal space. Ideally these formulae should not be stored but rather recalculated in an efficient way as they are needed.It is probably possible to use the new CASSCF CI method also in this context. C. C. J. Roothaan Rev. Mod. Phys. 1951 23 69; 1960,32 179. A. C. Wahl and G. Das in Methods of Electronic Structure Theory ed. H. F. Schaefer (Plenum Press New York 1977) vol. 3 p. 51. B. Levy Thesis (CNRS no. A0 5271 Paris 1971). S. F. Boys Proc. R. Soc. London Ser. A 1950 200 542. R. J. Buenker in Quantum Chemistry and Molecular Physics into the 80s,ed. P. G. 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