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LOCALIZED ORBITALS IN CHEMISTRY BY JOHN CHRISTOPHER CULBERSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA TN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987 ACKNOWLEDGEMENT I would like to thank my parents for their support and guidance throughout my life. Mary Kay your inspiration, infinite patience and willingness to wait kept me going. Thanks go to Bill Luken who taught me the basics.in quantum chemistry as well as introducing me to the computer as a learning tool. In addition, Bill Luken gave me an insight into the academic world. Finally, I would like to thank Bill and Marge for being our friends. I would like to thank Michael C. Zerner for allowing me to use the skills I learned at Duke and teaching me more quantum chemistry. The freedom he gave me to explore some of my own ideas as well as being guided occasionally was deeply appreciated. The entire Zerner family made our time at QTP enjoyable. Thanks go to my German host Dr. Notker Rosch for allowing me to come to Germany. My thanks to Peter and Monica Knappe for helping Mary Kay and me during our entire stay in Germany. We would like to thank Frau Brown for making us feel at home. One benefit of being a graduate student at the Quantum Theory Project is the wide variety of people you meet. One of the most enlightening experiences was to meet and take classes from Dr. N. Y. Ohrn. Thank you for giving me a new perspective on quantum chemistry. I would like thank to G. D. Purvis III for allowing me to help in designing the C3D program and giving me plenty of experience debugging/expanding the INDO code once a day. Your persistence in asking the question "Well why do you want to do that?" help me formulate problems more completely. No graduate student can ever learn about life in a large research program without a great postdoc to help him or her along. Dan Edwards gave me a handle, provided constant assistance, and is a friend to talk to. It has been great to be a member of QTP and share in the wealth of experiences common only to QTP. The Sanibel symposium provided a chance to meet some of the most unique people in the word. I would like to thank all of the members of QTP, especially the secretarial staff, for making my stay here great. Last but not least thanks to the boys and girls of the clubhouse. Thanks go to Bill reminding me that learning something does not have to be boring. Thanks go to Charlie reminding me that you don't understand something until you can explain it to someone else. Thanks go to Alan showing me that some theory can still be done on a piece of paper. All of the members of the clubhouse have provided me with an atmosphere conducive to the free exchange of ideas on quantum theory and everything else. TABLE OF CONTENTS Page ACKNOWLEDGMENT........................................... ii LIST OF TABLES........................................... v LIST OF FIGURES.......................................... vi ABSTRACT................................................ vii INTRODUCTION............................................. 1 CHAPTER ONE LOCALIZED ORBITALS................... 3 Background ........................... 3 Double Projector Localization........ 6 Fermi Localization................... 9 Boys Localization.................... 34 CHAPTER TWO LANTHANIDE CHEMISTRY................. 41 Background........................... 41 Model................................ 43 Procedures............................ 61 Results.............................. 62 CONCLUSION............................................... 71 BIBLIOGRAPHY............................................. 73 BIOGRAPHICAL SKETCH...................................... 79 LIST OF TABLES Page 11 Probe electron points for furanone............. 18 12 Boys and Fermi hole centroids for C4H402....... 19 13 Probe electron points for methlyactetylene..... 22 14 Boys and Fermi hole centroids for CHCH3........ 23 15 Orbital centroids for BF3...................... 26 16 Eigenvalues and derivatives for BF3 using the Boys method .................................... 27 17 Probe electron points for BF3.................. 28 18 Orbital centroids for BF3...................... 29 21 Basis functions for Lanthanide atoms............ 51 22 Ionization potentials for Lanthanide atoms..... 54 23 Average configuration energy for Lanthanides... 57 24 Resonance integrals for Lanthanide atoms....... 58 25 Geometry and ionization potentials for Cerium and Lutetium trihalides......................... 63 26 Geometry and ionization potentials for Lanthanide trichlorides........................ 65 27 Geometry and ionization potentials for SmC12, EuCl2 and YbC 2......................... .. 67 2 28 Geometry of Ce(NO3) 6 ion...................... 68 29 Population analysis of Ce(N03 62 ion........... 70 3 6 LIST OF FIGURES Page 11 Fermi mobility function for H2CO................ 12 12 Difference between mobility function and electron gas correction......................... 13 13 Fermi hole plot for formaldehyde................ 14 14 Boys localized orbital for formaldehyde......... 15 15 Ni(CO)4 bonding orbital......................... 38 16 Ni(CO)4 nonbonding orbital..................... 39 17 Ni(CO) antibonding orbital..................... 40 21 Single and double C basis set plot.............. 49 22 Average value of r versus atomic number......... 50 23 Pluto plot of Ce(NO 2......................... 69 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LOCALIZED ORBITALS IN CHEMISTRY by John Christopher Culberson May 1987 Chairman : Michael C. Zerner Major Department : Chemistry The localized orbitals discussed here will be divided into two classes: (1) intrinsically localized orbitals, where the localization is due primarily to symmetry or energy considerations, for example transition metal dorbitals or lanthanide forbitals; and (2) orbitals which must be localized after a selfconsistent field (SCF) calculation. In the latter case, two new methods of localization, the Fermi and the double projector methods, are presented here. The Fermi method provides a means for the noniterative localization of SCF orbitals, while the double projector allows one to describe what atomic functions the localized orbitals will contain. The third localization procedure described is the second order Boys method of Leonard and Luken. This method is used to explain the photodissociation products of Ni(CO)4. The Intermediate Neglect of Differential Overlap (INDO) method is extended to the forbitals, and the intrinsic localization of the f orbitals is examined. This extension is characterized by a basis set obtained from relativistic DiracFock atomic calculations, and the inclusion of all onecenter twoelectron integrals. Applications of this method to the lanthanide halides and the twelve coordinate 2 Ce(NO3)62 ion are presented. The model is also used to calculate the ionization potentials for the above compounds. Due to the localized nature of forbitals the crystal field splitting in these compounds are extremely small, leading to SCF convergence problems which are addressed here. Even when the SCF has converged, a small configuration interaction (CI) calculation must be done to insure that the converged state is indeed the lowest energy state. The localized nature of the f orbitals in conjunction with the double projector localization method may be used to isolate the forbitals in order to calculate only a CI restricted within the fmanifold. viii INTRODUCTION Localized orbitals may be defined as either orbitals which are spatially compact or as molecular orbitals which are dominated by a single atomic orbital. The use of the terms bond, antibond, or lone pair to describe a set of orbitals are all based on a localized orbital framework. The use of ball and stick models and hybrid orbitals in every general chemistry class illustrates the power of localized orbitals as an aid in the understanding of molecular structure. Localized orbitals may be divided into two categories. The first category encompasses orbitals which must be localized. Although, some orbitals are localized automatically either by their symmetry or by their energy relation to other orbitals in the molecule, these orbitals form the basis for the second category of localized orbitals. Transition metal dorbitals fall into this second category, and it has been predicted that the lanthanide forbitals should also fall into this category. Our understanding of transition metal chemistry is also based on the concept of orbitals being localized. The excitations that give rise to the colors of many metal complexes are classified as dd, ligandd or charge transfer. These classifications are based on the fact the d orbitals are localized allowing for the easy interpretation of spectra. The success of crystal field theory reinforces the belief that the d orbitals are localized. In the following two chapters, I will examine both of these types of localized orbitals. The first chapter will deal with methods developed to obtain localized orbitals from delocalized orbitals, and the use of 2 such methods on several systems of chemical interest. Chapter two deals with expanding the INDO method so that the prediction of forbitals being localized orbitals may be verified and so that the unique bonding and spectroscopy of these compounds may be examined. By adapting the INDO method, we may now expand our studies to include the chemistry of the lanthanides and actinides. The chemistry of the lanthanides and actinides is different from the chemistry of the corresponding dorbital chemistry. The compact (localized) nature of the forbitals, causes the ff spectral transitions to be characterized by very sharp transitions and the positions of the transitions are almost unaffected by the ligands attached to the metal. The forbitals are potentially involved in expanding the valence of lanthanide containing compounds; some lanthanide molecules have a coordination number of nine and several twelve coordinate lanthanide compounds are known. Are forbitals required for greater valency, or is the greater valency merely a consequence of the larger ionic radius of most lanthanides? The study by quantum chemical methods has been slowed by the size of the lanthanide containing molecules, but the INDO method lends itself to the study of large molecules and therefore the choice was made to expand the INDO model to include forbitals. CHAPTER ONE LOCALIZED ORBITALS Background The observable properties of any wavefunction composed of a single Slater determinant are invariant to a unitary transformation of the orbitals occupied in the wavefunction.1 Because of this invariance, the observable properties of a closedshell selfconsistent field (SCF) wavefunction may be described using canonical orbitals, or any set of orbitals related to the canonical orbitals by a unitary transformation. Canonical orbitals are quite useful in postHarteeFock calculations for several reasons. Canonical molecular orbitals (CMOs) are obtained directly by matrix diagonalization from the SCF procedure itself. The canonical orbitals form irreducible representations of the molecular point group. Since the symmetry is maintained, all subsequent calculations may be simplified by the use of symmetry. Spectroscopic selection rules are determined using the canonical orbitals. Koopman's theorem, which relates orbital energies to molecular ionization potentials and electron affinities, is based entirely on the use of canonical orbitals. Localized orbitals (LMOs) allow for the wavefunction to be interpreted in terms of bond orbitals, lonepair orbitals and inner shell orbitals, consistent with the Lewis structures learned in freshmen chemistry. Unlike CMOs, LMOs may be transferred into other wavefunctions as an initial gu ss, thereby reducing the effort needed to produce wavefunctions for large molecules. The most important use of 4 localized orbitals is their ability to simplify configuration interaction (CI) calculations. LMOs maximize intraorbital electronic interactions and minimize interorbital electronic interactions. This concentrates correlation energy into several large portions instead of many small portions as given by the CMOs. A major disadvantage of the use of localized orbitals is the loss of molecular point group symmetry. Localized orbitals do not transform as an irreducible representation of the molecular point group. The total wavefunction, of course, does. Localization methods may be divided into several categories. The first category of localization is based on an implicit definition of what a localized orbital should be. An underlying physical basis for localized orbitals is exploited in the second category of localization. Localized orbitals may also be produced in accord with the users own definition of localization. The implicit definition on which the localized orbitals are produced differs from method to method but all of these methods proceed in a similar fashion. A function of the form n G = E i=l ii is maximized or minimized, where the definition of on the localization criterion. One choice for the value of 23 choice the sum G is maximized. This method is referred to as the Edmiston Ruedenberg (ER) method. Perhaps the most popular choice of a localization method is the Boys method, in which the g operator is the orbital selfextension operator,6 gii = 2 (12) 12 5 This form of the g operator may be recast in terms of the product of two molecular orbital dipole operators. One can relate this form of localization to maximizing the distance between the orbital centroids. Once a localization criterion has been established, the next step is to construct a transformation matrix to do the localization. Since the exact nature of the transformation is unknown, an iterative procedure is used to construct the localized orbitals. This iterative procedure moves from a lesslocalized set of orbitals to a morelocalized set. Once a convergence criterion is met i.e., the orbitals do not change within a given tolerance, the iterative procedure is stopped. Although localizations using either of the above two methods are relatively standard, some problems may be encountered. As with any iterative procedure, convergence difficulties may be encountered. In the case of the ER method all twoelectron repulsion integrals must be transformed on each iteration, a very time consuming step proportional to N5. Since the SCF procedure itself proceeds as N (semiempirical) or N4 (ab initio) and the systems studied here are large, we will not consider the ER method of localization any further. The same integral transformation problem is encountered for the Boys method, but since the integrals involved are dipole (oneelectron) integrals the problem is much simpler. Since the localization criteria are so different there is no reason to expect different methods to yield orbitals that are similar, but in general the LMOs are quite similar for the Boys and ER methods. These orbital similarities lead to the :;scond category of localization. We claim that the underlying physiciJ b'as.i of onrdlization is the Fermi hole. The Fermi hnolo provide : n a dir(t (nloi.tei rtive) method 6 for transforming canonical orbitals to localized orbitals. The integral transformations that limit the usefulness of the Boys and ER methods are also eliminated when using this method. The disadvantage of this method is the fact that a series of probe points must be generated for the molecule. These points may be generated using chemical intuition or by 12 a search of the Fermi hole mobility function.2 The Fermi hole method of localization may also fall into the final category since it can be made to pick out a particular localized orbital set. The final category of localization method allows one to produce orbitals in accordance with one's needs. As mentioned above, the Fermi method may be classified in this category, but another method was developed especially for this purpose, one that we have called the double projector (DP) method. This method has been used in conjunction with the other methods above to help predict the lowest energy state of lanthanide containing compounds where forbital degeneracies are a problem. The DP method allows one to separate the forbitals from the other metal orbitals and use a small CI to determine the ground state of the molecule. Double Projector The double projector (DP) method of localization is an extremely useful method for localizing orbitals when the form of the localized orbitals is known or suspected in advance. For example, if one would like to study nn transitions in a molecule, a full localization need not be done, the double projector may be used to isolate (localize) the ntype orbitals. A subsequent small singles CI may then be used to study only the nn transitions and thereby elucidate the nn spectra. 7 Another example involves the localization of the dorbitals in a transition metal complex. Because of accidental degeneracies between metal dorbitals and ligand molecular orbitals (MOs), the atomic d orbitals may be spread out in many canonical orbitals. A large CI is then required to restore the localized nature of the d type molecular orbitals. Such a large CI can be avoided by first doing a DP localization. The DP method can also be used to remove orbitals from the orbital set so that the remaining orbitals may be localized using a standard localization technique. For example, a common problem with a Boys localization is the mixing of a and R orbitals to obtain T orbitals, this is not desirable since the a and n spectra will now be mixed and more difficult to interpret. The n orbitals may be removed using the DP method, the remaining orbitals localized, and the nn spectra calculated using a small singles CI. The double projector is a complementary method of localization and is normally used in conjunction with other traditional methods of localization; therefore, no examples of its use will be given here. An outline of the double projector method is given in this section. Consider a set of m occupied spin orbitals ([im, and a set of r localized "pattern" orbitals (['i}, where r is less than or equal to m. These "pattern" orbitals are projected out of the set ('i}l by m IT,> = E I.X>< IT > (13) a i=l i for a = 1 to r. These (I'>}1 are then symmetrically orthogonalized '+' = A (14) S= 61/2 (15) and are projected out of the original set {i}, = ( ~ l > The matrix A' is formed and diagonalized U V 'U = U+A U = X (17) The X matrix will have r near zero eigenvalues corresponding to the (8 )1 that have been projected out. These eigenvalues and the corresponding columns of U are removed. The new set of orthonormalized orbitals (Y }m"r is formed from oc 1 Y = 4'UX1/2 (18) This set is an orthogonal complement to the set je >, but has no particular physical significance. To obtain a set of orbitals most like the canonical set, we form F, the Fock matrix, over the Y subset and diagonalize F, +Y +FYV = er (19) Y' = YW (110) Y' are linear combinations of Y that we can energy order according to e These Y' orbitals are the most like the original canonical orbitals with the "pattern" orbitals removed. 9 Fermi Localization Background This section presents a method for transforming a set of canonical SCF orbitals into a set of localized orbitals based on the properties of the Fermi hole711 and the Fermi orbital.13'14 Unlike localization methods based on iterative optimization of some criterion of localization,26,15,16 the method presented here provides a direct (non iterative) calculation of the localized orbital transformation matrix. Consequently, this method avoids the convergence problems which are possible with iterative transformations. Unlike the extrinsic methods for transforming canonical SCF orbitals 1719 into localized orbitals,1719 the method presented here does not depend on the introduction of a definition of a set of "atomic orbitals". The method presented here may also be distinguished from applications of localized orbitals such as the PCLIO method2025 in that the latter method does not involve SCF orbitals, and it is not concerned with the transformation of canonical SCF orbitals into localized orbitals. Properties of the Fermi Hole The Fermi hole is defined as A(rl;r2) = p(rl) 2p2(rl,r2)/p(r2), (111) where p(rl) is the diagonal portion of the first order reduced density matrix and p(rl;r2) is the corresponding part of the second order reduced density matrix.26 For special case of a closed shell SCF 10 wavefunction, the natural representation of the Fermi hole is the absolute square of the Fermi orbital13,14 A(rl;r2) = If(rl;r2)2. (112) The Fermi orbital is given by 1/2 f(rl;r2) = 2/p(r2)]1/2 gi(rl)g(r2), (113) i where the orbitals gi(r) are either the canonical SCF molecular orbitals or any set related to the canonical SCF molecular orbitals by a unitary transformation. The Fermi orbital f(rl;r2) is interpreted as a function of r1 which is parametrically dependent upon the position of a probe electron located at r2. Previous work 12,13,27,28 has demonstrated that the Fermi hole does not follow the probe electron in a uniform manner. Instead, molecules are found to possess regions where the Fermi hole is insensitive to the position of the probe electron. As the probe electron passes through one of these regions, the Fermi hole remains nearly stationary with respect to the nuclei. These regions are separated by regions where the Fermi hole is very sensitive to the position of the probe electron. As the probe electron passes through one of these regions, the Fermi hole changes rapidly from one stable form to another. The sensitivity of the Fermi hole to the position of the probe electron is measured by the Fermi hole mobility function,12,27,28 F(r) = Fx(r) + F (r) + Fz(r) (114) where F(r) =2 j j 2 (115) S2 avj av i>j for v = x, y or z. This may be compared to FO(P) = (3i/4)(p/2)2/3 (116) which provides an estimate of the Fermi hole in a uniform density electron gas. The Fermi hole mobility function F(r) for the formaldehyde molecule is shown in Fig. 11. The difference F(r)F0(p) is shown in Fig. 12. Regions where F(r) > F(p) that is, the Fermi hole is less sensitive to the position of the probe electron than it would be in an electron gas of the same density, may be compared to the loges proposed by 2933 Daudel. Regions where F(r) = F(p) resemble boundaries between loges. When the probe electron is located in a region where F(r) < F(p), the Fermi orbital is found to resemble a localized orbital determined by conventional methods.26,15,16 This similarity is demonstrated by Figs. 13 and 14 which compare a Fermi hole for the formaldehyde molecule with a localized orbital determined by the orbital centroid criterion of localization.46,15 Localized Orbitals Based on the Fermi hole Equation 113 provides a direct relationship between a set of canonical SCF orbitals gi(r) and a localized orbital f.(r) = f(r,r.) where r. is a point in a region where F(r.) < F0(p(rj)). In order to transform a set of N canonical SCF orbitals into a set of N localized orbitals, it is necessary to select N points r. j = 1 to N, each of which is located in a region where F(rj) < F (p(r.)). Ideally, each of these points should correspond to a minimum of F(r) or F(r)FO(p). This condition, however, is not critical, because the Fermi hole is relatively insensitive to the position of the probe electron when the probe electron is located in one of these regions. Figure 11: The fermi hole mobility function F(r) for the H CO based on the geometry and double zeta basis set of ref. 41. The locations of the nuclei are indicated by (+) signs. The contours represent mobility function values of 0.1, 0.25, 0.5, 1.0, 2.0 and 5.0 atomic units. The contours increase from 0.1 near the corners, to over 5.0 in regions enclosing the carbon and oxygen nuclei. Each nucleus is located at a local minimum of the mobility function. / Figure 12: \ / II S(+ / ' S\ \ \I The difference between the Fermi hole mobility function F(r) and the electron gas approximation for the H CO molecule. The contours represent values of 0.0, 0.1, 0.25, 0.5, 1.0, 2.0 and 5.0, in addition to those indicated in figure 11. the contours representing negative values and zero are indicated by broken lines. Each nucleus is located at a local minimum. Figure 13: The fermi hole for the formaldehyde molecule determined by a probe electron located at one of the protons. The contours indicate electron density of 0.005, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28 and 2.56 electrons per cubic bohr. (D + Figure 14: The localized orbital for the CH bond of a formaldehyde molecule determined by the orbital centriod criterion for localization. The electronic density contours are the same as in figure 13. 16 A set of N Fermi orbitals determined by Eq. 13 is not generally orthonormal. Each member of this set, however, is usually very similar to one member of an orthonormal set of conventional localized orbitals. Consequently, the overlap between a pair of Fermi orbitals is usually very small, and a set of N Fermi orbitals may easily be converted into an orthonormal set of localized orbitals by means of the method of symmetric orthogonalization.33 The resulting unitary transformation is given by U = (TT+)1/2T, (117) where 1/2 T. = g.(r.)/(p(r.)/2) (118) J3 3 1 1 In the following three sections, the transformation of canonical SCF orbitals based on Eqs. 117 and 118 is demonstrated for each of three molecules. The first example, a cyclic conjugated enone, represents a simple case where conventional methods are not expected to have any special difficulties. The second example, methyl acetylene, is a molecule for which conventional methods have serious convergence problems.34 The third example, boron trifluoride, is a pathological case for the orbital centroid criterion, with a number of local maxima and saddle points in the potential surface according to the Boys criterion of localization. In each case, the first step in the application of this method is the selection of the set N points. This set always includes the locations of all of the nuclei in the molecule. For atoms other than hydrogen, the resulting Fermi orbitals are similar to innershell localized orbitals. When the probe electron is located on a hydrogen atom, the Fermi orbital is similar to an RH bond orbital. 17 Additional points for the probe electron may usually be determined based on the molecular geometry. The midpoint between two bonded atoms (other than hydrogen) tends to yield a Fermi orbital resembling a single bond. Multiple bonds may be represented with two or three points located roughly one to two bohr from a point midway between the multiply bonded atoms, along lines perpendicular to a line joining the nuclei. Likewise, lone pair orbitals may be determined by points located roughly one bohr from the nucleus of an atom which is expected to possess lone pair orbitals. Application to the Furanone Molecule The furanone molecule, C4H402, and its derivatives are useful 3537 reagents in 2+2 photochemical cycloadditions. 7 The canonical SCF molecular orbital for the furanone molecule were calculated with an STO 3G38 basis set and the geometry specified in Table 11. The molecular geometry was restricted to Cs symmetry, with a planar five membered ring. Fermi hole localized orbitals were calculated based on the set of points indicated in Table 11. These points include the positions of the ten nuclei, as well as twelve additional points determined by the method outlined above. The centroids of the localized orbitals determined by the points in Table 11 are shown in Table 12. The C=C and C=O double bonds are each represented by a pair of equivalent bent (banana) bonds similar to those determined by other methods for transforming canonical SCF orbitals into localized orbitals. As shown in Table 12, the centroids of the orbitals determined by the Fermi hole are very close to those of the localized orbitals Page Table 11 : Molecular geometry and probe electron points for the furanone (C4H402). The first ten points indicate the molecular geometry used in these calculations. The twelve additional probe electron positions were determined as described in the text. All coordinates are given in bohr. Position X Y Z Atom C1 0.0 0.0 0.0 Atom C2 0.0 2.589 0.0 Atom C3 2.671 3.355 0.0 Atom C4 4.363 0.940 0.0 Atom 01 2.534 1.071 0.0 Atom 02 3.427 5.548 0.0 Atom H1 1.802 1.040 0.0 Atom H2 1.802 3.629 0.0 Atom H3 5.560 0.858 1.698 Atom H4 5.560 0.858 1.698 C1C2 bond 1 0.0 1.295 2.000 C1C2 bond 2 0.0 1.295 2.000 C2C3 bond 1.336 2.972 0.0 C3C bond 3.517 2.148 0.0 C401 bond 3.449 0.066 0.0 C101 bond 1.267 0.536 0.0 C302 bond 1 3.049 4.452 2.000 C302 bond 2 3.049 4.452 2.000 01 lone pair 1 2.654 1.821 0.660 01 lone pair 2 2.654 1.821 0.660 02 lone pair 1 2.707 6.298 0.0 02 lone pair 2 4.267 5.698 0.0 Table 12 : 19 Orbital centroids for localized orbitals determined by the Fermi hole method and by the orbital centroid method for the furanone molecule (C4H402). All coordinates are given in bohr. Fermi hole method Centroid criterion Orbital X Y Z X Y Z C1 K shell 0.0 0.001 0.0 0.0 0.0 0.0 C2 K shell 0.0 2.588 0.0 0.0 2.588 0.0 C3 K shell 2.670 3.355 U.0 2.671 3.355 0.0 C4 K shell 4.362 0.940 0.0 4.326 0.939 0.0 01 K shell 2.534 1.070 0.0 2.533 1.070 0.0 02 K shell 3.427 5.547 0.0 3.426 5.547 0.0 C1H1 bond 1.138 0.742 0.0 1.171 0.761 0.0 C2H2 bond 1.148 3.318 0.0 1.181 3.331 0.0 C4H3 bond 5.149 0.895 1.132 5.155 0.888 1.153 C4H4 bond 5.149 0.895 1.132 5.155 0.888 1.153 C1C2 bond 1 0.008 1.401 0.635 0.030 1.410 0.599 C1C2 bond 2 0.008 1.401 0.635 0.030 1.410 0.599 C2C3 bond 1.320 3.054 0.0 1.284 3.046 0.0 C3C4 bond 3.542 2.190 0.0 3.565 2.149 0.0 C401 bond 3.268 0.243 0.0 3.243 0.225 0.0 C101 bond 1.472 0.627 0.0 1.495 0.613 0.0 C302 bond 1 3.093 4.584 0.548 3.114 4.612 0.511 C302 bond 2 3.093 4.584 0.548 3.114 4.612 0.511 01 lone pair 1 2.554 1.283 0.440 2.594 1.311 0.472 01 lone pair 2 2.554 1.283 0.440 2.594 1.311 0.472. 02 lone pair 1 3.015 5.886 0.0 3.011 5.892 0.0 3.970 02 lone pair 2 5.589 0.0 3.970 5.600 0.0 determined by the orbital centroid criterion.46,15 Likewise, the localized orbitals determined by the Fermi hole were found to be very close to those determined by the orbital centroid criterion. Each of the localized orbitals determined by the Fermi hole method was found to have an overlap of 0.994 to 0.999 with one of the localized orbitals determined by the orbital centroid criterion. The remaining (off diagonal) overlap integrals between these two sets of localized orbitals were found to have a root mean square (RMS) value of 0.011734. The transformation of a set of canonical SCF orbitals to an orthonormal set of localized orbitals determined by the Fermi hole required 10 minutes on a PDP11/44 computer. The orbital centroid (Boys) method required 140 minutes starting from the canonical SCF molecular orbitals or 80 minutes, using the Fermi localized orbitals as 5 an initial guess, to reach TRMS of less than 10 where TRMS is the RMS value of the oEfdiagonal part of the transformation matrix which converts the orbitals obtained on one iteration to those of the next iteration. The orbital centroid criterion calculations reported here are based on a partially quadratic procedure which requires less time than conventional localization procedures based on 2X2 rotation. Application to Methylacetylene The localized orbitals of methylacetylene are of interest because of the convergence difficulties encountered in attempts to calculate these orbitals using iterative localization methods. These difficulties are caused by the weak dependence of the criterion of localization on the orientation of the three equivalent CC (banana) bonds relative to the three CH bonds of the methyl group. In calculations based on the 21 orbital centroid criterion, over 200 iterations were required to determine a set of orbitals which satisfied a very weak criterion of 34 convergence. Most of these difficulties may be overcome using the 15 quadratically convergent method which has been developed recently. As shown below, however, the localized orbitals based on the Fermi hole yield nearly equivalent results and require much less effort than even the quadratically convergent method. The canonical SCF molecular orbitals for methylacetylene were determined by an STO5G basis set38 and an experimental geometry.39 Transformation of the 11 occupied SCF orbitals into localized orbitals based on the Fermi hole method required the selection of 11 points. These points are shown in Table 13. The positions of the nuclei provided seven of these points. One point was located at the midpoint of the C2C3 single bond. The remaining three points were located two bohr from the I' rotation axis at a point midway between the C1 and C2 nuclei. These last three points were eclipsed with respect to the methyl protons. The centroids of the localized orbitals determined by this method are shown in Table 14. As expected, the triple bond is represented with three equivalent banana bonds. The centroids of the corresponding orbitals determined by the orbital centroid criterion are also shown in Table 14. These are very close to those determined by the Fermi hole method. The RMS value of the offdiagonal part of the overlap matrix between the localized orbitals determined by the Fermi hole and those determined by the orbital centroid criterion is 0.012874. The Fermi hole method required 1.62 minutes to transform the canonical SCF molecular orbitals into an orthonormal set of localized molecular orbitals. By comparison, the (quadratically convergent) Page Tablp 13 : Molecular geometry and probe electron positions for the methylacetylene molecule. The first seven points indicate the locations of the nuclei. All coordinates are given in bohr. Position I Y Z Atom C1 0.0 1.140 0.0 Atom C2 0.0 1.140 0.0 Atom C3 0.0 3.897 0.0 Atom H1 0.0 3.145 0.0 Atom H2 1.961 4.438 0.0 Atom H3 0.980 4.438 1.698 Atom H4 0.980 4.438 1.698 C1C2 bond 1 2.0 0.0 0.0 C1C2 bond 2 1.0 0.0 1.732 C1C2 bond 3 1.0 0.0 1.732 C2C3 bond 0.0 2.510 0.0 Table 14 : 23 Orbital centroids for localized orbitals determined by the Fermi hole method and by the orbital centroil method for the methylacetylene molecule. All coordinates are given in bohr. Fermi hole method Centroid criterion Orbital X Y Z X Y Z C1 K shell 0.0 1.141 0.0 0.0 1.140 0.0 C2 K shell 0.0 1.139 0.0 0.0 1.134 0.0 C3 K shell 0.0 3.896 0.0 0.0 3.896 0.0 C1H1 bond 0.0 2.486 0.0 0.0 2.502 0.0 C3H2 bond 1.302 4.270 0.0 3.316 4.272 0.0 C3H3 bond 0.651 4.270 1.127 0.658 4.272 1.139 C3H4 bond 0.651 4.270 1.127 0.658 4.272 1.139 C1C2 bond 1 0.713 0.003 0.0 0.692 0.013 0.0 CiC2 bond 2 0.356 0.003 0.617 0.346 0.013 0.599 C1C2 bond 3 0.356 0.003 0.617 0.346 0.013 0.599 C2C3 bond 0.0 2.457 0.0 0.0 2.498 0.0 24 orbital centroid method required 26 minutes starting from the canonical SCF molecular orbitals or 19 minutes using the Fermi localized orbitals 8 as an initial guess to reach TRMS of 108 or less. Application to Boron Trifluoride As a further example of the application of the Fermi hole localization method, localized orbitals were also calculated for the boron trifluoride molecule. The molecule provides a demonstration of how characteristics of little or no physical significance can cause serious convergence difficulties for iterative localization methods. A localized representation of boron trifluoride includes four innershell orbitals, three boron fluorine bond orbitals, and nine fluorine lone pair orbitals. The orbital centroid criterion method shows a small dependence on rotation of each set of three lone pair orbitals about the corresponding BF axis. Consequently, the hessian matrix for he criterion of localization as a function of a unitary transformation of the orbitals has three very small eigenvalues. The optimal orientation of the fluorine lone pairs may correspond to one of several possible conformations. One of these, the "pinwheel" conformation, has a single lone pair orbital on one of the fluorine atoms in the plane of the molecule. The other two lone pair orbitals on this fluorine atom are related to the first lone pair by 120 degree rotations about the FB axis. The lone pair orbitals on the other two fluorines are obtained by 120 degree rotations about the C3 axis. The point group of the orbital centroids for this conformation is C3h. The "threeup" conformation is generated by rotating the set of lone pair orbitals on each fluorine atom in the pinwheel conformation by 90 25 degrees about each FB axis. The point group for the orbital centroids of this conformation is C3v. The "upupdown" conformation is generated by rotating the set of lone pair orbitals on one of the fluorine atoms in the "threeup" conformation by 180 degrees about the FB axis. This conformation has the symmetry of the Cs point group. The canonical SCF orbitals for BF3 were calculated based on the doublezeta basis set and geometry tabulated by Snyder and Basch.40 Localized orbitals determined by the orbital centroid criterion were obtained for the threeup conformation and the upupdown conformation. The centroids for these orbitals are shown in Table 15. The first five (most positive) eigenvalues of the hessian matrix for each of these conformations are shown in Table 16. All eigenvalues of the hessian matrix are negative for both of these conformations, indicating that both conformations are maxima for the sum of squares of the orbital centroids. The pinwheel conformation, however, was never found. Consequently, it was not possible to exclude the possibility that the pinwheel conformation was the global maximum and the threeup conformation was only a local maximum. The pinwheel conformation can easily be constructed using the Fermi hole localization method by selecting an appropriate set of probe positions. This set of points is shown in Table 17. The centroids of the resulting set of localized orbitals are shown in Table 18. When this set of localized orbitals is used as the starting point, the orbital centroid method quickly converges to a stationary point with C3h symmetry. The centroids of the resulting set of orbitals are shown in Table 18. As shown in Table 16, three of the eigenvalues of the hessian matrix were positive at this point, demonstrating that the Page Table 15 : 26 Orbital centroids for localized orbitals determined by the orbital centroid method for the boron trifluoride molecule. The four innershell orbitals have been excluded from these calculations. All coordinates are given in bohr. upupup (C3v) upupdown (Cs) Orbital X Y Z X Y Z BF1 bond 1.725 0.0 0.072 1.725 0.0 0.070 BF2 bond 0.862 1.494 0.072 0.862 1.494 0.070 BF3 bond 0.862 1.494 0.072 0.862 1.494 0.069 F1 lone pair 1 2.445 0.0 0.487 2.446 0.0 0.487 lone pair 2 2.596 0.433 0.207 2.596 0.433 0.208 lone pair 3 2.596 0.433 0.207 2.596 0.433 0.207 F2 lone pair 1 1.222 2.117 0.487 1.222 2.118 0.487 lone pair 2 0.922 2.465 0.207 0.922 2.465 0.208 lone pair 3 1.674 2.032 0.207 1.674 2.031 0.207 F3 lone pair 1 1.222 2.117 0.487 1.223 2.119 0.486 lone pair 2 0.922 2.465 0.207 0.922 2.464 0.208 lone pair 3 1.674 2.032 0.207 1.673 2.031 0.208 Table 16 : Values of the orbital centroid criterion and the second derivatives of the orbital centroid criterion for various conformations of localized orbitals for the boron trifluoride molecule. The row labelled sum indicated the sum of the squares of the orbital centriods for each of the conformations. The following rows show the five highest (most positive) eigenvalues X. of the corresponding hessian matrix. The first and second columns correspond to the localized orbitals described in described in Table 15. The third and fourth columns correspond to localized orbitals described in Table 18. The gradient vectors are zero for the first three columns. Configurations upupup upup down pinwheel Fermi hole Sum 69.445360 69.444985 69.438724 69.341094 X1 0.019521 0.038174 +0.017234 +0.031444 X2 0.019850 0.018914 +0.016238 +0.029932 X 0.019850 0.019095 +0.016236 +0.029928 X4 0.155480 0.153827 0.193521 0.195502 X5 0.155484 0.155181 0.1n4366 0.195524 Table 17 : Probe electron positions for the boron trifluoride molecule. The first three points are located at the midpoint of the BF bonds. The remaining points have been chosen in the pinwheel conformation (symmetry C3h). All coordinates are given in bohr. Position X Y Z BF1 bond 1.223 0.0 0.0 BF2 bond 0.611 1.059 0.0 BF3 bond 0.611 1.059 0.0 F1 lone pair 1 2.781 0.943 0.0 lone pair 2 2.718 0.472 0.817 lone pair 3 2.781 0.472 0.817 F2 lone pair 1 2.207 1.937 0.0 lone pair 2 0.981 2.644 0.817 lone pair 3 0.981 2.644 0.817 F3 lone pair 1 0.573 2.880 0.0 lone pair 2 1.799 2.172 0.817 lone pair 3 1.799 2.172 0.817 Table 18 : 29 Orbital centroids for localized orbitals determined by the Fermi hole method and by the orbital c:ntroid method for the boron LrifluoLide molecule. The ufur innershell orbitals have been excluded from these calculations. All coordinates are given in bohr. Fermi hole Centroid criterion Orbital X Y Z X Y Z BF1 bond 1.713 0.0 0.0 1.720 0.028 0.0 BF2 bond 0.857 1.484 0.0 0.884 1.475 0.0 BF3 bond 0.857 1.484 0.0 0.835 1.504 0.0 F1 lonp pair 1 2.567 0.488 0.0 2.603 0.484 0.0 lone pair 2 2.538 0.244 0.414 2.520 0.256 0.410 lone pair 3 2.538 0.244 0.414 2.520 0.2',6 0.410 F2 lone pair 1 1.706 1.976 0.0 1.721 2.012 0.0 lone pair 2 1.057 2.320 0.414 1.037 2.310 0.410 lone pair 3 1.057 2.320 0.414 1.037 2.310 0.410 F3 lone pair 1 0.861 2.467 0.0 0.882 2.497 0.0 lone pair 2 1.480 2.076 0.414 1.482 2.054 0.410 lone pair 3 1.480 2.076 0.414 1.482 2.054 0.410 30 pinwheel conformation is a saddle point with respect to the orbital centroid criterion. These calculations also indicate that the threeup conformation is probably the global maximum for the orbital centroid criterion. We do not intend to attribute any special physical significance to any of the lone pair configurations for BF3. These calculations demonstrate some of the problems, such as local maxima and saddle points, which may occur for conventional iterative localization methods. These calculations demonstrate how the Fermi hole method may be used by itself to transform the canonical SCF oLbitals into localized orbitals without any of these difficulties. In addition, these calculations demonstrate how the Fermi hole method may be used in conjunction with the orbital centroid method to establish a characteristic of the orbital centroid criterion which would have been very difficult to establish using the orbital centroid method alone. Conclusions The numerical results presented here demonstrate how the properties of the Fermi hole may be used to transform canonical SCF molecular orbitals into a set of localized SCF molecular orbitals. Except for the symmetric orthogonalization, this method requires no integrals and no iterative transformations. The localized orbitals obtained from this method are very similar to the localized orbitals determined by the orbital centroid criterion. The orbitals determined by the Fermi hole may be used directly in subsequent calculations requiring localized orbitals. Alternatively, the orbitals determined by this method may be used as a starting point for iterative localization procedures.261516 31 The necessity of providing the set of probe electron positions may appear to introduce a subjective element into the localized orbitals determined by the Fermi hole method. Most of the subjective character to this choice, however, is eliminated by the fact that Fermi hole is relatively insensitive to the location of the probe electron whenever the probe electron is located in a region associated with a strongly localized orbital. This is reflected by the fact that the centroids of the localized orbitals determined by the Fermi hole method, as shown in Tables 12, 14 and 18, are much closer to the centroids of the corresponding localized orbitals determined by the orbital centroid criterion than they are to the probe electron positions used to calculate them. If the electrons are not strongly localized in certain portions of a molecule, such as in the lone pairs of a fluorine atom, then the Fermi hole may be more strongly dependent on the location of the probe electron than where the electrons are strongly localized. In such cases, the localized orbitals determined by the Fermi hole method may reflect the locations of the probe electron points more strongly than they are reflected in well localized regions. In such regions, .however, there may be no physically meaningful way to.distinguish between the localized orbitals determined by this method and those determined by any other method. In these situations, the Fermi hole method may provide a practical method for avoiding the convergence problems which may be expected for iterative methods when the electrons are not well localized. The electronic structure of most common stable molecules may be described by an obvious set of chemical bonds, lone pair orbitals, and innershell atomic orbitals. This is reflected in the success of methods such as molecular mechanics41'42 for predicting the geometries of complex molecules. The localized orbitals of such molecules are unlikely to be the objects of much interest in themselves, but they may be useful in the calculation of other properties of a molecule, such as 43 44,45 the correlation energy,43 spectroscopic constants, 5 and other properties.447 The selection of a set of probe electron positions for one of these molecules is simple and unambiguous, and the method presented here has significant practical advantages compared to alternative methods for transforming canonical SCF molecular orbitals into localized molecular orbitals. For some molecules, the pattern of bonding may not be unique or it may not be entirely obvious, even when the geometry is known. For example, two or more alternative (resonance) structures may be involved in the electronic structure of such molecules. The localized orbitals of such molecules may be of interest in themselves, in order to characterize the electronic structure of such molecules, in addition to their utility in subsequent calculations.4347 In order to apply the current method to such molecules, the Fermi hole mobility function228 must be used to resolve any ambiguities which may arise in the selection of the probe electron positions. If two or more bonding schemes are possible, the positions of the probe electrons should be chosen to provide the minimum values of the Fermi hole mobility functions F(r) or the mobility function difference F(r)FO(p). In the case of methylacetylene, for example, the CC single bond may be determined by a single point midway between the carbon atoms, where F(r) is less than FO(p). Any attempt to represent this portion of the molecule with a double bond would require placing a probe electron away from the CC axis, in a region where F(r) is greater than FO(p). 33 Consequently, it is not possible to represent methylacetylene with a structure like HC=C=CH3 without placing one or more probe electron points in regions where the Fermi hole is unstable. In extreme cases, even the Fermi hole mobility function may fail to provide unambiguous positions for the probe electrons. This is expected in highly conjugated aromatic molecules, metallic conductors, and other highly delocalized systems. For these electrons, the method presented here, as well as all other methods for calculating localized orbitals, are entirely arbitrary. The electronic structure of such a delocalized system may be represented by an unlimited number of localized descriptions, each of which is equally valid. If there is a need for imposing a localized description on a highly delocalized system, the current method would be no less arbitrary than existing alternatives. The arbitrariness of the current method would be manifested in the choice of the probe electron positions for the delocalized electrons. However, the current method would continue to provide practical advantages over alternative transformations. These advantages include the absence of integrals to evaluate, the absence of iteratively repeated calculations, and the absence of convergence problems. Boys Localization The most widely used form of localized orbitals are those orbitals based on the method of Boys.5'6 The integral transformation procedure in any localization procedure can be a time limiting step. In the EdmistonReudenberg method the two electron repulsion integrals must be transformed an N5 computational step, but the Boys method may be 34 formulated in terms of products of molecular dipole integrals. The dipole integrals are oneelectron integrals therefore the transformation is on the order of N3, making the Boys localization the method of choice. One disadvantage of this method is that it is an iterative method which may be prone to convergence difficulties. Methods of Boys Localization There exists a unitaLy transformation relating a delocalized set of orbitals to a localized set, but the form of this transformation is in general unknown. The common method fol solving this problem is to do a series of unitary transformations that increase the degree of localization of a set of orbitals. Given a set of orbitals f one can increase the localization by doing a unitary transformation U , fl+1 = Uf1 (119) where f is the resulting set of more localized orbitals. The original UI matrix formulation is based on a sequence of pairwise 2 rotations, as proposed by Edmiston and Reudenberg. In this procedure, N orbitals are localized by rotating a pair of orbitals, then a second pair of orbitals is rotated, ., etc. until all N(N1)/2 pairs of orbitals have been rotated. Since the rotations are done in a specific order, the localized orbitals obtained will be dependent on the order of rotations. Leonard and Luken have developed a second order method that does all of the N(N1)/2 rotations at once rather than one at a time.5 The use of a second order method may have the additional benefit of improving the convergence difficulties encountered in iterative methods. Their method is outlined below. 35 The NxN unitary transformation matrix for the localization can be written as U = WR, (120) where V is a positive definite matrix defined by V = (RR)/2. (121) The matrix R will be defined as R = NT, (122) where T = 1 + t (123) and t is an antisymmetric matrix t+ = t (124) The N matrix is a diagonal matrix which normalizes the columns of R. By application of U to a set of orbitals (fl' fN) one produces a set of more localized orbitals (fi, fN}. The new value of the localization, G', is given by N G' = (i'i',i'i') (125) i'=l The new f' orbitals can be thought of in terms of a pertubative expansion, f' = f + Z t f + Z tt jf (126) I I,J to second order. The tI matrix does 2x2 rotations that mix in portions of all the occupied orbitals into orbital f'. The third term titJ is the product of a pair of 2x2 rotations. When this form of the f' 36 orbitals is substituted into the G' equation, you obtain G' = GO + Z t G1(I) + E tit G2(I,J) .(127) I Leonard and Luken5 include the G2 second order term to accelerate convergence when one is in the quadratic region; only the first order term G1 is calculated initially, and until the quadratic region is encountered. In practice, standard procedures for localization often can take several hundred iterations to converge. These second order procedure described above seldom takes more than 20 cycles, and the energy of orbitals related by symmetry (CH bonds in benzene, etc.) is usually reproduced to seven significant figures. Results One example of the Boys method of Leonard and Luken was given in the Fermi hole method section, in this section we will show the localized orbitals foi the Ni(CO)+ ion. In a recent experimental paper by Reutt et al., the photoelectron spectroscopy of Ni(CO)4is reported in order to clarify the nature of the transition metal carbonyl bond. Since the spectrum is interpreted in terms of localized orbitals on both the metal and the carbonyl groups, a first step in any quantum chemical treatment of the problem is to localize the orbitals. This system is somewhat complicated because it has an unpaired electron. The three T2 orbitals are occupied by five electrons leading to a triply degenerate ground state. The MOs are localized using the Leonard and Luken implementation of the Boys' procedure. The localization breaks the orbitals into several classes; (1) oxygen lone pairs, (2) carbon oxygen x (banana) bonds, (3) nickel carbon bonds, and 37 (4) nickel dorbitals of two types namely E and T2 type orbitals. The localization may also be done on the unoccupied orbitals; this separates the unoccupied orbitals into two sets (1) nickel carbon antibonds and (2) carbon oxygen antibonds. In Fig. 15, an isovalue plot of one of the nickel carbon bonds is shown. The orbital shows a large amplitude near the carbon atom, indicative of large porbital contributions on the carbon and a relatively small contribution from the nickel dorbitals. As one can see from Fig. 16, a sizable contribution to bonding comes from one of the partially occupied nickel dorbitals. The nickel carbon antibond shown in Fig. 17 possesses a large node along the internuclear axis. The nickel carbon bond is expected to be quite weak, because it is composed of a sum of the two bonding orbitals shown in Figs. 15 and 16. The diffuse nature of the photoelectron spectra indicates the population of additional vibrational modes, resulting from a distortion from a tetrahedral geometry. The weak nickel carbon bonds would allow for such a distortion to take place in the ion. A further application of the Boys method would be to include the localized orbitals into a limited CI calculation to see if one could predict the photoelectron spectra for Ni(CO)4. The use of any localized orbital technique does not add or subtract information from the overall wavefunction. These methods only divide orbitals into more chemical pieces allowing for easier interpretation of experimental results. 0 C \ ! C 0 Figure 15: An isovalue localized orbital plot of a nickel carbon bonding orbital in the Ni(CO)+ molecule. The dashed lines indicate an orbital amplitude of 0.05 a.u. per cubic bohr. The solid lines indicate an orbital amplitude of 0.05 a.u. per bohr. 0 C o' OC A C Figure 16: An isovalue localized orbital plot of a nickel carbon non bonding orbital in the Ni(CO)+ molecule. The dashed lines indicate an orbital amplitude of 0.05 a.u. per cubic bohr. The solid lines indicate an orbital amplitude of 0.05 a.u. per bohr. 0 II C \ O Figure 17: An isovalue localized orbital plot of a nickel carbon anti bonding orbital in the Ni(CO), molecule. The dashed lines indicate an orbital amplitude of 0.05 a.u. per cubic bohr. The solid lines indicate an orbital amplitude of 0.05 a.u. per bohr. CHAPTER TWO LANTHANIDE CHEMISTRY Background The past decade has seen a dramatic increase in interest and activity in lanthanide and actinide chemistry. Not only has considerable knowledge been gained in the traditional area of inorganic felement chemistry, but much modern work is concerned with organof 49 element reactions,49 and the use of lanthanides and actinides as very specific catalysts.551 Unlike the corresponding chemistry involving the d metals, very little explanation is offered for much of this chemistry. The electronic structure of these systems is difficult to calculate from quantum chemical means for several reasons. Most of the complexes of real experimental interest are large. In addition, veiy little about forbitals as valence orbitals is known, although experience is now being gained on the use of f orbitals as polarization orbitals. Finally, the forbital elements are sufficiently heavy that relativistic effects become important. Very few ab initio molecular orbital studies 52 have been reported on forbital systems. Extended Huckel calculation, however, have been successful in explaining some of this chemistry.53 Scattered wave and DVM Xa studies of forbital systems have also proven effective, especially in examining the photoelectron spectroscopy of reasonably complex systems. 56 We examine an Intermediate Neglect of Differential Overlaps (INDO) technique for use in calculating properties of forbital complexes. At the SelfConsistent Field (SCF) level this technique executes as rapidly 41 II on a computer as does the Extended Huckel method, and considerably more rapid than the scattered wave Xac method. Since the electrostatics of the INDO method are realistically represented, molecular geometries can be obtained using gradient methods.57 Since the INDO method we examine contains all onecenter twoelectron terms it is also capable of yielding the energies of various spin states in three systems. With configuration interaction (CI) this model should also be useful in examining the UVvisible spectra of forbital complexes. Preliminary studies of forbital chemistry using an INDO model have been disclosed by Clack and Warren58 and, more recently, by LiMin, JingQuing, Guang 59 Xian and Xiu Zhen.59 The method we examine will differ from their methodology in several areas, as discussed below. Several problems unique to an INDO treatment of these systems must be considered, and we have very little ab initio work to guide us. As mentioned, what role do relativistic effects play? Although we might hope to parameterize scalar contributions through the choice of orbitals and pseudopotential parameters, spin orbit coupling, often larger than crystal field effects, will need to be considered at some later stage. Since forbitals are generally tight, and ligand field splitting thus small, a great many states differing only in their forbital populations lie very close in energy. These near degeneracies often prevent "automatic" SCF convergence, a problem with which we must deal for an effective model. The nature of the valence basis set itself is in question. Are the filled 5p and the vacant 6p of the lanthanides both required for a proper description of their compounds? Model The INDO model Hamiltonian that we use was first disclosed by Pople and collaborators,60 and then adjusted for spectroscopy61 and extended to the transition metal series.6264 The details of this model are published elsewhere.626 To extend this model Hamiltonian to the f orbital systems we need first a basis set that characterizes the valence atomic orbitals, and that is subsequently used for calculating the overlap and the one and twocenter twoelectron integrals. Subsequent atomic parameters that enter the model are the valence state ionization potentials used for calculating onecenter oneelectron "core" integrals and the SlaterCondon Fk and Gk integral that are used for thr formation of onecenter twoelectron integrals. The evaluation of these integrals using experimental information has traditionally made 65,66 this model highly successful in predicting optical properties.6 We ,mploy in this model one set of pure parameters, the resonance or B(k) parameters; for each lanthanide atom we decided to use B(s) = B(p), B(d) and B(f). These parameters will be chosen to give satisfactory geometries of model systems. Another choice is one that gives good prediction: of UVvisible spectroscopy.61,63 These values seldom differ much from those chosen to reproduce molecular geometry. In this initial work all twocenter twoelectron integrals required for the INDO model Hamiltonian are calculated over the chosen basis set, as are the onecenter twoelectron F integrals. An alternate choice would be one that focuses on molecular spectroscopy. In such a case, and one that we have to investigate subsequently, the onecenter two electron F could be chosen from the Pariser approximation67 FO(n) = IP(n) EA(n), (IP = Ionization Potential, EA = Electron Affinity) and the twoelectron twocenter integral from one of the more successful function established for this purpose.6870 functions establishedd for this purpose. 44 At the SCF level, we seek solutions to the pseudoeigenvalue problem F C = C (21) with F, the Fock or energy matrix, C, the matrix compound of Molecular Orbital (MO) coefficients, and s, a diagonal matrix of MO eigenvalues. The above equation is for the closed shell case (all electrons paired). The uncestiicted Hartree Fock case is discussed in detail elsewhere,61 7173 as is the open :shell restricted case.7173 Although nearly all f orbital systems are open shell, consideration of the closed shell case demonstrates the required theory and is considerably simpler. Within the INDO model, elements of F are given by: F AA= U + PX[ lax (a (l) (22a) [a,X] CA + E P ( iir a) E ZB (liulsBB) acB B#A *A A FA = PaX [(I l) (avX) u.t (22b) FAB 1[ F = A(p) + BB(v) S 2 P' ('iVv) APB (22c) where rd 1 * (iv u ) = Jdr(1)d.(2) Xr(1) X (l) r12 X(2) XX(2) (23) P is the first order density matrix, and since one assumes that the Atomic Orbital (AO) basis {X} is orthonormal it is identical to the charge and bond order matrix, given by MO P = E C Ca na (24) a AB with n the occupation of MO a n = 0,1,2. In Eq. (22), FB refers to a matrix element integral UAA ( 1 2 (25) is essentially an atomic term and will be estimated from spectroscopic data as described below. V is an effective potential that keeps the valence orbital X orthogonal and noninteracting from the neglected innershell orbitals. The choice of an empirical procedure for UA will remove the necessity for explicit consideration of this term. The bar over an orbital in an integral, AA such as (0 ii indicates that the orbital X is to be replaced with an s symmetry orbital of the same quantum number and exponent. The appearance of such orbitals in the theory is required for rotational symmetry and compensates for not including other two center integrals of the NDDO type;74 i.e. (u A A, X#X. The last term in Eq. (22a) represents the attraction between an electron in distribution X* X and all nluclei but A. The rationale for replacing integral AA (U IR1 uA)  (p S B) (26) is given elsewhere, and compensates for neglected two center inner 27,28 shellvalence shell repulsion228 and neglected valence orbital (symmetrical) orthogonalization. 29 is the core charge of atom B 46 and is equal to the number of electrons of neutral atom B that are explicitly considered; i.e. 4 for carbon, 8 for iron, 4 for cerium, etc. 1315 S of Eq. (22c) is related to the overlap matrix D, 5 and is given by SU = E f (l)V(1) g () ( ((v(1)) (27) 1=0 where gu(1)v(1) is the Eulerian transformation factor required to rotate from the local diatomic system to the molecular system, (u(l)jv(l)) are the sigma (1=0), pi(l=l), delta(l=2) or phi (1=3) components to the overlap in the local system, and fu(l)v(l) are empirical weighting factors chosen to best reproduce the molecular orbital energy spread for model ab initio calculations. We have made little use of this f factor, and set all f = 1 except between p symmetry orbitals viz.61 S = 1.267gp (palp) + 0.585p (pnpg) (28) r P + 0.585gppE (pn Ip ) Basis Set In general ZDO methods choose a basis set of Slater Type Orbitals (STO) R = 1 rnl Cr Y1 (89,) (29a) nim [2n! e where Y,(9,O) are the real, normalized spherical harmonics. Atomic orbitals XP are expressed as fixed contractions of these {Rnlm} X = E anlm Rnlm (29b) In general a single Rn1m function describes the s and p orbitals for most atoms. The d orbitals of the transition metals, however, require at least a doubleC type function (two terms in 29b) for an accurate description of both their inner and outer regions. For the lanthanides we have examined basis sets suggested by Li LeMin et al.,59 by Bender and Davidson,78 and by Clementi and Roetti.79 In the latter case, the two major contributors of Eq. (29b) in the valence orbitals of the doubleC atomic calculations were selected, and these functions were renormalized with fixed ratio to yield the required nodeless doublec functions for IDO. We were unable with any of these choices to develop a systematic model useful for predicting molecular geometries (see later discussion of resonance integrals). We have adapted the following procedure on selecting an effective 80 " basis set. Knappe and Rosch calculated the lanthanides and their monopositive ions using the numerical DiracFock relativistic atomic program of Desclaux.81 From these wavefunctions jadial expectation values functions. The 6s, 5d and 4f wavefunctions were obtained by DiracFock calculations on the promoted, 4fm35d16s2 configuration; the 6p from calculations in which a 5d electron was promoted, 4fm36s26p. Wavefunctions for the monopositive ions are obtained from 4,33d 6s and 4fm36':6pl respectively. A generalized Newton procedure was then used to determine exponents () and coefficients anlm for a given set of in the transition metal atoms, we found that a single function fits the ns and np atomic functions well in the regions where bonding is 48 important, but the (nl)d, and now the (n2)f require at least two terms in the expansion of Eq. (29b). This is demonstrated for the Ce+ ion in Figure 21, where it is shown that a singleC expansion is poor for the outer region of the 4f function. In Figure 22 the value of contraction of the 6s and 6p orbitals due to relativistic effects (DF vs. HF) is quite apparent here, and is a consequence of the the greater core penetration of these orbitals. Subsequent expansion of the 4f and 5d, now with increased shielding, results. After some experimentation we use the DiracFock values obtained from thf: monopositive ions. The basis set adopted is given in Table 21. The 4f and 5d functions are quite compact. At typical bonding 1 distance (4f4f iv) and (5d5dlpu) are essentially RAB. Because of this we calculate all twocenter twoelectron integrals with the C1 values in Table 21. This value is chosen to match the accurate e FO SlaterCondon Factors obtained from the numerical atomic calculations by a single exponent, via F(4f4f) = 0.200905 C (4f) (210a) F(5d5d) = 0.164761 (5d) (210b) F(6s6s) = 0.139803 C (6s) (210c) F(6p6p) = 0.139803 C (6p) (210d) The error in calculating twocentered twoelectron integrals at typical bonding distances with this singlec approximation is well under 17, and this procedure is much simpler. Core Integrals The average energy of a configuration of an atom or ion is given by82,83 Single vs. Double Zeta 41STO Orbital Amplitude 0.20 0.17 0.13 .,0.10 0.07 0.03 0.00 0. Double Zeta Single Zefta 1.5 2.0 2.5 3.0 3.5 4.0 4. 60 r in a.U. Figure 21: Radial wavefunction for the 4f orbital of Ce+ with singleC and doubleC Slater type orbitals (STOs). 0 0.6 1.0 * IA I I * */ \  '\ < \ % r .5 ~5% 5%5 I I DF vs. HF Average Values of r for the 4f,5d.6s and 6p Orbital Of 51 I 60 62 64 66 Atomic number 68 70 Dr6p A HF 8p a DF a x DP 5d * HW 6d + DI4 o HI 4 72 Figure 22: Average value of r for the valence orbitals of the lanthanides from a relativistic calculation (DF) and a non relativistic calculation (HF). II I . .  4U S0 w . 4,0 4 0 4.1 . u 0 0 C 0 0 0 a .4 .a1 4 0a 1 o 'o "o r * C0 A 4 4J 000 on C L0 U0 0 S**' a 3 CO 0 4 o4 A 4.1 0 A, 0 C 0 U4 .4 4 C E 4' T o cu M 'O& n O r N t r 4 r4 rf4 n . tn NJ 4. * 0 0' 0 fn ca C QC 0 4.1 4 _1 I 1 I 1 r f . . I I I . . I I I I r t r ) t r I I I I I I I I I 1 a 1 w 6 E sk mdq = k Uss+ mUpp + nUdd + qU + 1 W (Sss pp dd ff 2 ss (211) m(m1) + n(n) + q(l +kmW + knWs + 2 pp 2 dd 2 ff sp sd kqWsf + mnWpd + mqWVp + nqVdf with W.., the average two electron energy of a pair of electrons in IJ orbita ls Xi and Xj given by W ss PP Wdd Wff sp Wsd Wsf Wpd Wpf Wdf = F(ss) = FO(pp) = Fo(dd) = F(ff) = FO(sp) = FO(sd) = FO(sf) = F(pd) = FO(pf) = Fo(df) The core integrals (212) 2/25F2(pp) 2/63F2(dd) 2/63 F (dd) 4/195F2(ff) 2/143 F4(ff) 100/5577 F6(ff) 1/6G1(sp) 1/10 G2(sd) 1/14 G3(sf) 1/15 G1(pd) 3/70 G3(pd) 3/70 G2(pf) 2/63 G4(pf) 3/70 Gl(df) 2/105 G3(df) 5/231 G5(df) U.., Eq. (25), are then evaluated by removing an 11 electron from orbital Xi, and equating the difference in configuration energy between cation and neutral to the appropriate observed IP(n). We prefer this procedure rather than that suggested by others that average 77,82,84 the value obtained from IP(n) and EA(n).77 There are a great many low lying configurations of the lanthanide atoms and their ions. The lowest terms of Ce, Gd and Lu come from fn3d s 2, while the remaining lanthanide atoms have the structure fn2 s2 Two processes are then possible for 6s electron ionization: I fn3d1s2 fnd s + (s) II fn2 s2 fn2sl + (s) 53 The ionization energy of a 6s electron from I is systematically 0.4  0.5 eV larger than that obtained from II. When combined with Eq. (2 11), the estimate for Uss differ by less than 0.1 eV. That is, choosing the values of process I, the use of Eq. (211) predicts the values of process II within 0.1 eV. We thus choose the values of process I shown in Table 22. These values are obtained from the promotion energies of Brewer85, 86 and then smoothed by a quadratic fit throughout the series. For completeness, we also give the values of process II. The lowest configuration containing a 5d electron is fn3d s2 throughout the series, and 5d ionizations are obtained from III fn d1s2 fn3s2 + (d) The ionization potentials for the 6p can be obtained from two processes: IV fn2 slp1 4 nsl + (p) V fn3 s2p 3s2 + (p) Ionization from process IV is nearly constant at 3.9 eV, from V at 4.6 eV. The fn 2s2 configuration is lower for all the lanthanides except Ce(fdsp), and Gd and Tb(fn3s2p). Using il ionization potentials of process IV, and Eq. (211), we predict the values of process V to within 0.2 eV. We do not consider this error significant, and thus use the smoothed values from IV given in Table 22. The values from process V are also given in the table for comparison. For a f orbital ionization, we consider the two processes VI f,3 ds2 fn4ds2 + (f) VII fn2 s2 n3s2 + (f) (compare with I and II). As seen in Table 22 the values form the two processes are very different. From Eq. (211) U (VI) = IP(VI) (m4) W 2Wf Wdf (213a) U f (VII) = IP(VII) (m3) Wf 2Wsf (213b) 0 .4 a 41 0 I) 5r.  3 a a 0 U U4 41 0 0 o 4 a 4 So 0 ~ H N 0 In 14 r. I 00 .0 to 1u 0 0 In I I o S3 44 4 a 0 0400 40 C 0.4 4 * 404 m 41 U. * 0 N o C H O 4 In N X 1 In E<4 S>I o0 in In v w Ch w 0) 0% r N m 4 0 InN 'C a Mn 1 n %a rN N in W i 0% T wa en m C N rI co c a m 0 .4 .4 x u .14 ..I I S 04 0 v' CN in r. 'o ir '. v1 m '.4 m o 44 Ln %o rI 'o In Imp r4 N IC r1 o U . S 4 I " 0 r in 0 1 mn o 0% IC ( i ,0 0 M : H 'N 'I N m n Ln 10 In I 'C I' 0 I H H 1 In a 1 1N 0 ean ia in '0 0 0  4 .4 I 'o r qo 1) 0o N 0o 1. 0 'C i N a O 1 0%o N i^. .I In m M in o N n H 0 > 'C 0 o (A 0a C 0 0 0 i n o 'C N 0o n en 0p% r' In en M .4 0% r 'o I il .  I__ _ ITn o on no I 0 0 In 10 V a p0 e  41 % 0. 0 0. 0 w N H 0 ID 1 o4 O I I v o .0 >, 0 1 1 .0 : 41 0. Z* C. Ill Cyi l. u & 0 Q (. F. >1 l 55 Unlike the analogous situation for the 6s and 6p orbitals, use of Eq. (213a) to find Uff, and use of this value in Eq. (213b) to predict IP(VII) is not successful, and would require the scaling of the large Fo(ff) integral often performed in methods parameterized on molecular spectroscopy.616367 As with the transition metal nd orbitals we might envision the following procedure. We assume that the lanthanide atom in a molecule is a weakly perturbed atom. The lowest energy configuration of the atom should than be most important in determining Uff. We create a twoby two interaction matrix ( n32 2ds2 C = 0 (214a) S V E(f2s2) X C2) 2 2 where V is an empirical mixing parameter, and C1 and C22 determines the relative amounts of each of the two configurations that are important. The exact value of V would depend on a given molecular situation. C1 is then given by 2 C = (214b) 1 1+X2 X = C/C= E(f 3d s2) E(f"2s2) + (214c) E(f3d s2) E(f2s2 1 22V The values of C1 appear in Table 22, where we have used the values of E(fm3ds2) and E(f2s2) obtained for the promotion energies of Brewer and a fixed value of V = 0.02 au. Then Uff could be obtained from U = C12 ff(VI) + C22 U ff(VTI) (215) In the case of the 3d orbitals this valence bond mixing between 3dn2 s and 3d ns was important in obtaining reasonable geometric predictions,63 an observation now confirmed in careful ab initio 87 studies.8 For the lanthanide complexes of this study the 4f orbitals are quite compact, and this valence bond mixing does not greatly affect geometries. However, the calculation of ionization potentials that result in states with reduced forbitals occupation is influenced. There are many refinements one can make in the formation of a "mixing" matrix such as Eq. (214a). One might be to make V dependent on the calculated population of the 4f and 5d atomic orbitals. However, the values of the promotion energies we obtain from Brewer are so different than those that we obtain from out own numerical calculations on the average energy of a configuration, Table 23, that for the moment we choose a 76% : 24% mix of E(fm3ds2) : E(fm2s2) for all the atoms of the series. This mix gives reasonable geometries and ionization potentials for all molecules of this study. Further refinements will require more accurate atomic promotion energies and numerical experience with the model. Resonance Parameters, B(k) Each lanthanide atom has three B (k) values, B(s) = B(p), B(d) and B(f), and those we choose are summarized in Table 24. They are obtained by fitting the geometries of the trihalides, and the more covalent biscyclopentadienyls to be reported elsewhere. Bond lengths are most sensitive to B(d) and bond angles to B(p). These angles can be reproduced solely on a basis set including 6p orbitals, and we have been able to obtain satisfactory comparisons with Table 23 : Average configuration energy from DiracFock calculations on the fn3ds2 and the f" s2 configurations for all the lanthanide atoms. Average Configuration Atom Energy fn3d s fnms Ce 8853.71494569 8853.64980000 Pr 9230.41690970 9230.3/981848 Nd 9616.94751056 9616.93446923 Pm 10013.4526061 10013.4606378 Sm 10420.0710475 10420.0976615 Eu 10836.9533112 10836.8834715 Gd 11264.0945266 11264.0439334 Tb 11701.7877496 11701.7482691 Dy 12150.1565528 12150.1286785 Ho 12609.3663468 12609.3484161 Er 13079.5686394 13079.5585245 Tm 13560.9236801 13560.9201649 Yb 14053.5770354 14053.5786047 Lu 14557.7153258  a) Reference 81. 58 Table 24 :Resonance integrals (B values) for the Lanthanide atoms in e V. The beta for the sorbital is set equal to the beta for the porbital. Atom B(s) B(p) B(d) B(f) Ce 8.00 8.00 17.50 80.00 Pr 7.61 7.61 17.58 80.00 Nd 7.23 7.23 17.65 80.00 Pm 6.85 6.58 17.73 80.00 Sm 6.46 6.46 17.81 80.00 Eu 6.08 6.08 17.88 80.00 Gd 5.69 5.69 17.96 80.00 Tb 5.31 5.31 18.04 80.00 Dy 4.92 4.92 18.11 80.00 Ho 4.54 4.54 18.19 80.00 Er 4.15 4.15 18.27 80.00 Tm 3.77 3.77 18.35 80.00 Yb 3.38 3.38 18.42 80.00 Lu 3.00 3.00 18.50 80.00 59 experiment without the necessity of including the 5p orbitals. On the other hand, orbitals of p symmetry do seem to be required for accurate 58,59 predictions of geometry.5859 It has been argued that the 4f orbitals are not used in the chemical bonding of those complexes except in the more covalent cases.55'56 From the present study we are lead to the conclusion that some, albeit small, contribution is required of these orbitals to obtain the excellent agreement between experimental and calculated bond lengths for the series MF3, MCI3, MBr3 and MI3 and for the comparative values obtained for CeF3 and CeF4. This is indicated in Table 24 by the large values of IB(f) The latter values are a consequence of the fact that the f orbitals are tighter than one usually expects for orbitals important in chemical bonding. Use of 5d orbitals alone will predict the trends in these two series, but underestimates the range of values experimentally observed. Two Electron Integrals Several different interpretations have been given to the INDO scheme. The simplest of those schemes is to include only onecentered integrals of the Coulomb or exchange type (uulvv) or (uvlvu) For an s,p basis these are complete. For an s,p,d or s,p,d,f basis they are not, and the omission of the remaining integrals will lead to rotational variance. To restore rotational invariance, integrals of 88 this type might be rotationally averaged,88 but from a study of spectra it appears that all onecenter integrals should be evaluated.6 For example, in the metallocenes the integral (d 2 2 d xId d ) is required to separate the two transitions that arise from the elg(d) 4 e (d) transitions that lead to the Elg and E2g excited states. In 2g1g2 60 addition, it appears that the inclusion of all onecenter integrals improves the predictions of angles about atoms with s,p,d basis sets89,90 and considerably improves the predictions of angles about the lanthanides. For these reasons we include all the onecenter two electron integrals. Since the INDO programs we use process integrals and their labels in the MOLECULE Format91 only the additional integrals need he included. These integrals are generated in explicit form via a computer program that we have used in the past63 and they have also been recently published by Schulz et al.89 To our knowledge all these integrals do not appear in the literature for s,p,d and f basis, although we have checked those of (uujvv) and (uJv\uv) against the formulas of Fanning and Fitzpatrick.83 Integrals of the form (uujvv) and (uvlvu) can be obtained though atomic spectroscopy, and their components, Fk and Gk, evaluated via least square fits (i~ivv) = C ak Fk k (PIvlu) = T bk Gk k These Fk and Gk can then be used to evaluate all integrals of the "F" or "G" type, even those that do not appear in atomic spectra because of high symmetry (i.e. (d 2 2 d zdxz d )). Integrals of the "R" type, x y yz xz xy however, cannot be evaluated in this manner; viz.(sdldd), (splpd), (sdlpp), (sdlff), (sfldf), (pplpf), (ddlpf), (pdldf), (sdlpf), (pdlsf), (spldf), and (pflff). For this reason we evaluate all onecenter two electron integral of the lanthanides using the basis set of Table 21, which yields the exact FO value obtained from the FockDirac numerical calculations. All Fk, Gk and Rk integral for k > 0 are then scaled by 61 2/3. This value of the scaling is obtained from a comparison of the calculated and empirically obtained589294 F2(ff), F4(ff) and F(f) values that implies 0.66 + 0.04. Empirically obtained values of Gk(fd) and F (fd) are far more uncertain and are much smaller, and are thus not used to obtain this scaling value between calculated and experimental values. At this point it seems appropriate to point out the differences of the present INDO model to that suggested by Li LeMin et al.59 In the latter formalism only the conventional onecenter twoelectron integrals are included leading to rotational variance. In addition, the WolfsbergHelmholz approach is used for the resonance integral B, B..=(IP(i)+IP(j))S../2. No geometry optmization has been reported within their model.59 Further differences are the restriction to singlec STOs and the smoothing of the valence orbital ionization 59 potentials for the lanthanides via Annotype expressions.5 Procedures The input to the INDO program consists of molecular coordinates and atomic numbers. Molecular geometries are obtained automatically via a gradient driven quasiNewton update procedure,57 using either the restricted or unrestricted Hartree Fock formalism. All UHF calculations are followed by simple annihilation.62 Selfconsistent field convergence is a problem with many of these systems. For this reason electrons are assigned to molecular orbitals that are principally f in nature according to the number of felectrons in the system, and the symmetry of the system. Orbitals with large 62 lanthanide 5d character are sought and assigned no electrons. A procedure is then adopted that extrapolates a new density for a given Fock matrix based on a Mulliken population analysis of each SCF cycle.93 Often this procedure is not successful. In such cases all f orbitals are considered degenerate, and they are equally occupied in the 71 highest spin configuration using the RHF open shell method. These vectors (orbitals) are then stored, and the SCF repeated with the specific f orbital assignments as described above. In cases of slow convergence, a singles or small singles and doubles, CI is performed to check the stability of the SCF, and the appropriateness of the forced electron assignment to obtain the desired state.96 Results The geometries of CeC13 and LuC13 were used to determine an optimal set of resonance integrals and configurational mixing coefficients. No further fitting was performed, and thus the structures of all other compounds are "predictions". The resonance parameters for the other lanthanides were determined by interpolation from the values for Ce and Lu (see Table 24). The INDO optimized geometries as well as the remaining cerium and lutetium trihalides are listed in Table 25. In addition to the trihalides reported, the geometry of CeF4 is also listed in Table 25. One can see the agreement with experiment is good in all cases. The potential energy of the trihalides as a function of the out of plane angle is very flat. Although we have optimized all structures until the gradients are below 104 a.u./bohr, the angles are converged only to 30. We note, however, that all are predicted nonplanar, in agreement with experiments.54b,97,98 Geometry and ionization potentials for Cerium and Lutetium trihalides. Cerium tetrafluoride is also included in this table. The bond distances are given in angstromsg angles in degrees and IPs in eV. Experimental results are also shown where available. a) References 54b, 97 and 98. LuF3 from Ref 103. Estimated values for CeF3,CeI3, and b) The SCF calculation on the ion of LuF3 would not converge therefore no IP is reported. Table 25 : Bond Bond Ionization Distance Angle Potential Molecule INDO Exp. INDO Exp. INDO Exp. CeF3 2.204 2.180 106.8  8.4 8.0 CeCl3 2.570 2.569 115.6 111.6 10.0 9.8 CeBr3 2.668 2.722 115.8 115.0 9.6 9.5 CeI3 2.844 2.927 119.8  9.9  CeF4 2.099 2.040 109.5 109.5   LuF3 2.045 2.020 107.4  b 19.0 LuCl3 2.415 2.417 108.2 111.5 18.6 (17.4 18.7) LuBr3 2.528 2.561 108.6 114.0 17.8 (16.8 18.4) LuI3 2.726 2.771 115.6 114.5 17.7 (16.2 18.1) 64 The experimental range of the bond lengths from LnF3 to Lnl3 is greater than we calculate. Our predicted values for the trifluorides and trichlorides are in good agreement, while bond lengths for the tribromides and triiodides are too short. Since these are the more polarizable atoms it is possible that configuration interaction will have its largest affect on these systems. The calculated change in bond length of 0.11 A in going from CeF3 to CeF4 is also smaller than the 0.14 A observed. Ionization potentials (IPs) are also reported in Table 25. In all cases the INDO values fall within the experimental ranges. These values are calculated usinf the ASCF method, and only the first IP is calculated. Experimentally54b,99 these valued are somewhat uncertain, but they are split by both crystal field effects, and by the large spin orbit coupling not yet included in our calculations. However, the 54b latter interaction is treated implicitly in the DVM Xa calculations4 based on the Dirac equation. Therefore, the Xa result for the ionization potentials show better agreement with the experiment in this aspect, but it is quite remarkable that the present INDO approach is able to reproduce the experimental trend in the first IP of the series, CeX3, X = F, Cl, Br with a maximum value for the chloride, a feature 54b noticeably missing in the DVM Xa results.5 The initial success of the INDO model as implemented here lead us to calculate both geometries and IPs for the remaining lanthanide trichlorides. These results are shown in Table 26. The experimental geometries54b,9798 are very well reproduced by the INDO calculations. The INDO IPs reproduce the characteristic "W" pattern of the lanthanide atoms, and fall within the experimental ranges. 65 Table 26 : Geometries and Ionization Potentials (IPs) for the lanthanide trichlorides. Bond distances are reported in angstroms, bgnd angles in degrees and IPs in eV. Experimental results are also given where available. Bond Bond Ionization Distance Angle Potential Atom INDO Exp. INDO Exp. INDO Exp. Ce 2.570 2.569 115.6 111.6 10.0 9.8 Pr 2.566 2.553 108.5 110.8 11.8 (10.911.2) Nd 2.563 2.545 112.7  13.3 12.0 Pm 2.556  112.7  14.4  Sm 2.544  113.0  15.3 (13.717.0) Eu 2.532  113.2  16.4  Gd 2.514 2.489 110.0 113.0 17.7 (15.516.5) Tb 2.496 2.478 109.8 109.9 13.0 (13.020.5) Dy 2.479  110.1  14.3 (14.020.0) Ho 2.464 2.459 112.0 111.2 15.0 (15.520.0) Er 2.448  110.9  15.6 (11.516.0) Tm 2.430  108.5  15.9 (15.321.0) Yb 2.421  109.6  15.9 (15.521.0) Lu 2.415 2.417 108.2 111.5 18.6 (17.418.7) a) References 54b, 97 and 99. 66 To test the applicability of our model to lanthanide atoms not formally charged +3, we calculated the geometries and IPs for SmCl2, EuCl2 and YbCl2 molecules. The results are given in Table 27. The INDO model gives optimized geometries that are bent and in good 100 agreement with experimental results. We note that this bending is a result of a small amount of porbital hybridization. It is not necessary to invoke London type forces, and thus correlation, to explain this effect. 2 We chose Ce(N03)6 as our last example because it is one of the few known examples of a twelve coordinate metal. The optimized geometry is summarized in table 28 and a plot of the optimized geometry is shown as figure 23. As one can see from Table 28 INDO predicts a geometry that is in excellent agreement with the experimental crystal structure.1 Table 29 shows a population study of this complex. Although there is some forbital participation, it appears that this unusual twelve coordinate Th structure results from electrostatic forces between the ligands and the relatively large size of the Ce(IV) ion. Geometry and ionization potential for SmC12, EuCl2, and YbCl2. Bond distances are given in angstroms, bond angles in degrees, and ionization potentials in eV. Experimental resultsa are listed where available. a) Reference 100. Table 27 : Bond Bond Ionization Molecule Distance Angle Potential INDO Exp. INDO Exp. INDO Exp. SmCI2 2.584  143.3 130+15 5.3 EuCI,, 2.576 143.2 13515 6.6 YbCl2 2.400  120.2 12605 3.2 Table 28 : 2 Average bond distances and bond angles for Ce(N03) 2 ion INDO optimized geometry and the Xray crystal structure Distances are in angstroms and angles in degrees. The c subscript on the oxygen atoms denotes the that oxygen is bonded to the cerium and the n subscript signifies a non bonded oxygen. a) Reference 101. Geometric Parameter INDO Exp. r(Ce0 ) 2.554 2.508 r(NOc) 1.256 1.282 r(NOn) 1.237 1.235 O(0N0 ) 121.5 114.5 0(0 Ce0 ) 50.9 50.9 C c CE(N03)6. _2 Figure 23: Plot of the twelve coordinate Ce(NO3)2 ion. Nitrogens 2, 18 and 22 are above the plane of the paper, while nitrogens 6, 10 and 14 lie below the plane of the paper. Table 29 : 2 Population analysis of Ce(N03)6 The oxygen atoms that are coordinated to the cerium are indicated by 0 .The Vyberg bond index is also given. A Wyberg index of 1.00 is characteristic of a single bond. Atomic Spin Total Atom Orbital Population Density Valence s 0.20 0.00 p 0.30 0.00 Ce d 1.32 0.00 f 1.10 1.00 Net 1.08 1.00 4.80 N Net 0.59 0.00 3.78 0 Net 0.40 0.00 1.58 c 0 Net 0.48 0.00 1.80 Bond Wyberg Bond Index Ce 0 0.40 N 0 1.37 N 0 1.22 CONCLUSIONS We develop an Intermediate Neglect of Differential Overlap (INDO) method that includes the lanthanide elements. This method uses a basis set scaled to reproduce DiracFock numerical functions on the lanthanide monocations, and is characterized by the use of atomic ionization information for obtaining the onecenter onee]r'ctron terms, and including all of the twoelectron integrals. This latter refinement is required for accurate geometric predictions, some of which are represented here, and for accurate spectroscopic predictions, to be reported latter. We have applied this method to complexes of the lanthanide elements with the halogens. The geometries calculated for these complexes are in good agreement with experiment, when experimental values are available. The trihalides are calculated to be pyramidal in agreement with observation. The potential for the umbrella mode, however, is very flat. The dichlorides of Sm, Eu and Yb are all predicted to be bent even at the SCF level, again in agreement with experiment. This bending is caused by a small covalent mixing of ungerade 6p and 4f orbitals, and one need not invoke London forces to explain this observation. Again the potential for bending is very flat. Within this model, forbitals participation in the bonding of these ionic compound through covalent effects is small. Nevertheless f orbitals participation does contribute to the pyrimidal geometry of .the trihalides and the bent structure of the dihalides. 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Zerner and M. Hehenberger, Chem. Phys. Letters 62, 550 (1979). 96. J.C. Culberson and M.C. Zerner, in preparation. 97. K.S. Krasnov, G.V. Girichev, N.I. Giricheva, V.M. Petrov, E.Z. Zasorin, N.I, Popenko, Seventh Austin Symp. on Gasphase Molecular Structure, Austin Texas, p. 88 (1978). 98. N.I. Popenko, E.Z. Zasorin, V.P. Spiridonov and A.A. Ivanov, Inorg. Chim Acta 31, L371 (1978). 99. E.P.F. Lee, A.W. Potts and J.E. Bloor, Proc. R. Soc. Lond. A 381, 373 (1982). 78 100. C.V. DeKock, R.D. Wesley and D.D. Radtke, High Temp. Sci. 4, 41 (1972); I.R. Beattie, J.S. Ogden and R.S. Wyatt, J. Chem. Soc. Dalton Trans., 2343 (1983). 101. T.A. Beineke and J. Del Gaudio, Inorganic Chemistry 7, No. 4, 715 (1968). 102. K.S. Krasnov, N.I. Giricheva and G.V. Girichev, Zhurnal Strukturnoi Khimii 17, 667 (1976). BIOGRAPHICAL SKETCH Chris Culberson was born in Saint Petersburg, Florida. He graduated from St. Petersburg Catholic High School. He obtained a Bachelor of Science degree with honors in chemistry from Eckerd College. He is married to Mary Kay Terns. After graduating from Eckerd College, he went to Duke University to study quantum chemistry under the direction of W. L. Luken. At Duke, the major portion of his research was devoted to localized orbital methods. Two years later, he transferred to the University of Florida to continue his studies under Michael C. Zerner's guidance. In addition to the forbital chemistry detailed in this thesis, a major portion of his time at the University of Florida was spent exploring the use of electrostatic potentials (EPs) and examining biochemical problems using EPs. While at the University of Florida, he was given the chance to go to Germany. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / 1 / / Michael C. Zerner, Chairman Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. N. Yngve Ohrn Professor 6f Chemistry and Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Willis B. Person Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. William Weltner Jr. Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sabin This dissertation was submitted to the Graduate Faculty of the Department of Chemistry in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1986 Dean, Graduate School UNIVERSITY OF FLORIDA 3 1111111111111111111111262 08554 158811111111111111111111 3 1262 08554 1588 