J. Phys. Chem. 1992, 96, 6264-6269
6264
Estimation of Magnetic Exchange Coupling Constants in Bridged Dimer Complexes J. R. Hart, A. K. Rap@,* Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523
S. M. Gorun, and T. H. Upton*,+ Corporate Research-Science Laboratories, Exxon Research and Engineering, Annandale, New Jersey 08801 (Received: February 13, 1992)
The validity of the Noodleman-Davidson unrestricted HartreeFock (UHF) broken symmetry calculational scheme is tested by extensive ab initio configuration interaction (CI) calculations on three simple systems. The molecular systems studied in the present work include H-He-H, the simplest example where a three center, four electron interaction exists, and [H-F-HIwhere the effects on J of having a polarizable entity in the center are investigated. Finally, the molecule CI4Ti20is used to include oxygen and a transition metal center. We fmd that using the difference between the energies of a high spin restricted Hartree-Fock (RHF) wave function and a low spin broken symmetry UHF wave function to calculate J agrees well with the large CI results. The magnetic coupling constants for H-He-H calculated from each of the different wave functions all give approximately the same results. The results of the full configuration interaction calculation are reproduced well at the restricted Hartree-Fock level. Having a fluorine atom between the two hydrogens changes the picture dramatically. The large CI results yield a J value significantly more negative than the RHF calculations. The projected JRu approximates the CI results very well over the range of singlet-triplet splittings usually considered to correspond to magnetic interactions (11OOO cm-I). In the transition metal system, Ti2C140,the effects of ligand polarization among the chlorine orbitals are neglected in the CI wave function due to calculational limitations. In order to treat the same orbital space in the UHF broken symmetry wave function, the chlorine orbitals are frozen at the RHF level. Good agreement between JRuand Jcl is achieved. For all three molecules, JRUcompares favorably to Jcr over a wide range of iron-bridge separations. Superexchange, direct exchange, and ligand spin polarization have been shown to be included in the broken symmetry approach. These contributions plus others are included in our CI calculations, but the computation time involved is much greater than the self-consistent field (SCF) calculations. The broken symmetry offers an adequate approximation for these effects at a much lower cost.
Introduction The theoretical characterization of the magnetic interactions of transition metal atoms in molecules and solids has been the subject of considerable recent interest. In bioinorganic chemistry, development of an understanding of the magnetic interactions between metal centers provides useful information about the coordination environment about the metal centers, provides information about the nature of bridging ligands, and provides an assessment about the distance between the metal centers and the potential for cooperative reactivity involving both centers. In solid-state inorganic chemistry (magnetism, high T, superconductivity), there is a general interest in developing a model of the distance dependence of the bridging ligand mediated spin coupling of electrons. The most common framework for discussing magnetic interactions in the literature is the model derived by Heisenberg, Dirac, and Van Vleck (HDVV).' The metal centers are assumed to be ions in a local ligand field, and the spins are assumed to interact isotropically to account for the magnetic exchange. While the HDVV spin Hamiltonian provides an accurate description of exchange interactions, it contains no information about the origin of these interactions, nor does it provide a systematic procedure for the accurate calculation and/or prediction of magnetic coupling constants for bridged transition metal molecules. Attempts at quantitative calculation of the magnetic coupling constants for bridged transition metal dimers that consider only the interactions between the metal ions have been shown to be inadequate. Contributions of orbitals on the bridging species beyond the Hartree-Fock level must be taken into account. More elaborate calculations incorporating direct bridge effects into the calculation of the coupling constants have been performed.2 De Loth and co-workers have implemented an approach using Rayleigh-Schrainger perturbation theory to obtain a direct expression for the energy difference between two spin states.2ad Calculations using perturbation theory on bridged copper d9-d9 dimers show that relaxing the orthogonality of the metal orbitals
'
Products Research Division, Exxon Research and Engineering, Linden, NJ 07036.
alone does not reproduce the experimental results. In fact, the magnetic exchange constant at this level of theory can be of the wrong sign; Le., the system is predicted to be ferromagnetic. Including the effects of the bridge lowers the singlet relative to the triplet enough so that the coupling constant is of the correct sign and order of magnitude. With this method, they attempted to incorporate many of the contributions proposed earlier into their approach, and the results were in qualitative agreement with those obtained experimentally. However, this type of perturbation calculation has only been applied to systems with one magnetic orbital per transition metal ion. As the number of electrons per center increases, so does the size of the calculation. In bridged d5 iron(II1) systems, for example, the size of a perturbation or (alternatively) a configuration interaction (CI) calculation necessary to describe the important bridge interactions becomes unmanageable. In order to avoid this difficulty, Xa or local density functional (LDF) theory, based on unrestricted Hartree-Fock (UHF) wave functions, has been used for a number of sulfur bridged iron systems? Noodleman and Davidson4have argued that the energy of a broken symmetry wave function obtained via a U H F formalism contains the important contributions to the coupling constant, inoluding the bridge effect called ligand spin polarization. The authors relate the difference between the energy of the highest spin state and that of the broken symmetry wave function to the coupling constant. This broken symmetry approach3 can be readily applied to systems with more than one magnetic electron per center though a consistent discussion of ground and several excited states is not possible with this methodology. In order to test the validity of the Noodleman-Davidson calculational scheme, we choose a few simple systems on which we can perform calculations at a high level of theory. We compare the results of using the Noodleman-Davidson4 scheme to full or nearly full CI wave functions. We determine the range over which the U H F broken symmetry approach can provide an approximation for the CI results. Three molecular systems are used in the present work. The first molecule is H-He-H, the simplest example where a three center, four electron interaction exists. Then [H-F-HI- is in-
0022-365419212096-6264%03.00/0 @ 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6265
Estimation of Magnetic Exchange Coupling Constants vestigated to test the effects of J of having a polarizable entity in the center. Finally, the molecule C14Ti20is used to include oxygen and a transition metal. We find that using the difference between the energies of a high spin R H F wave function and a low spin broken symmetry UHF wave function to calculate J agrees well with the large CI results.
Renew Only a brief outline of the Noodleman-Davidson analysis is included here. For the complete derivation, the reader is directed to the original paper.4 Noodleman and Davidson begin with a single normalized determinant which is the lowest energy solution to the open shell R H F equations:
Do = 14l(.1.....)l1)(.81(~~)~888...)
(1)
where d, represents n singly occupied d orbitals on the left with spin a,d, represents n singly occupied d orbitals on the right with spin 8, and 1 is the set of k doubly occupied orbitals. To construct the total wave function, determinants D, which are single or double excitations from Do, are defined. The normalized spin projected functions used for the total wave function are
where h’ is the nonlocal effective one electron operator:
(djaJh’)d,‘a)= ( d i a i h i c a ) + ~ : ( d i a + , i d d ; a+~ (dia+,igw;a) (10) where h is the local one electron operator. Ligand spin polarization, which is also a second-order energy, results from the fact that spin density appears on the ligands for all spin states. It can be represented by single ligand to metal excitations (lk@ di). The energy for such excitations is given by Slater’s rules as
-
Ef.dP(S) = I(lkglhlIdig)+ ( - l ) { [ n - S ( S - 1)]/n~}C(1~d,’ldd,‘di)1~
+
i
E-
Gj-G
kj
I(lkalhlldpa)+ (-l){[n - S(S - l)]/n2)E(lkdl~~~d14dp)12 9
E-
Eo,,-%
kp
where is the spin projection operator and the normalization constants have the form of squares of Clebsch43ordan coefficients. The total wave function @(S)is constructed from these normalized spin projected functions:
@(SI = *o(S) + Cau*u(S) U
(3)
where a, are the mixing coefficients for the first-order correction to 0:
(*,,ifw0)= - ( e-G
The effect of
DA = {d’l){aaa...){ 1v1k”){~~){d’r){&3/3...) = Do + CU,D,
( 1 2)
U
s ~ ~ u ~ ~1s ~ ~ o ) (4) -G (NaIY2
e
on Do is
‘00, = No[Do + CXoY(S)oY,]
(5)
V
where Xo,is the coefficient resulting from interchanging one spin index. It is obtained from Clebsch4ordan coefficients and is
XOY(S) = -
n-S(S+ 1) n2
It is now possible to express the first-order energy as a function of spin. The zero-order energy is E(O) = (Doll.rlDo), and the first-order energy is calculated using the relationship E(O) E(’)
= (*olW*o):
(11) Now that the first-order (eq 7), second-order superexchange (eq 9), and second-order ligand spin polarization (eq 1 1 ) perturbation energies have been expressed, connection to a broken symmetry wave function can be made. The broken symmetry UHF wave function can be expanded to first order in terms of the open shell R H F wave function:
+
where D,, are restricted to single excitations from Do and a, are determined without prior spin projection. That is, the broken symmetry wave function is assumed to be made up of all possible spin states between S = 0 and S = S,,,. The first-order energy for the broken symmetry wave function Ek’) can be written as snu
E t ) = CAI(S)[n- S(S + ~ ) ] J F S
(13)
where AI(S) = No@),which is the weighting factor for spin state S, and JFis defined as the ferromagnetic part of the coupling constant:
JF = C (djd,’lgld,‘di) / n Z ij
(14)
Using eqs 7 and 13, the first-order energy difference between the state with the highest spin and the broken symmetry wave function is therefore
- E t ) = -Sma:JF (15) where the following spin coefficient identities were used: where g = l / r 1 2 . The second-order energies can also be expressed. Superexchange, or Anderson’s kinetic exchange, arises from single excitations in Do from metal to metal. The most general form of this energy is
% .m lx
C A l ( S ) S ( S+ 1 ) = n = S,,,
s-0
(16)
The second-order energy of the broken symmetry wave function due to superexchange is given as
which, upon substitution of the normalization constants, can be written as
6266 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
Upon rearrangement, a manageable form for the second-order energy is obtained E&
= CA,(S)[n(n+ 1) - S(S + l)lJsE S
(19)
where JSE
=
(20) For superexchange, the relationship between the energy of the highest spin state and the energy of the broken symmetry wave function is given from eqs 9 and 18 as E&(Smax)
- E & E = -Sma?JSE
(21)
The second-order energy of ligand spin polarization is also determined:
After much mathematical manipulation, the result is
E~&P= cAl(S)lCL~p + [ n - S(S + ~)IIJLsP (23) S
where Cupis a term common to all spin states. It is of the form
Hart et al.
Geometries md Basis Sets We begin with the H-He-H system because it is the simplest case where there is a three center, four electron interaction. Several H-He bond distances between 1.25 and 5.0 A are used, with the H-He-H angle held at 180'. The basis set for helium3 is a (6s/3s) contracted basis set optimized for atomic He with a single p Gaussian with an exponent of 1.27 added. The hydrogen basis5is also a (6s/3s) plus p (exponent = 0.6)for a total eighteen basis functions for the H-He-H system. For the [H-F-HI- ion, the same basis is used for hydrogen, and the Dunning6 (9s5p/3s2p) contraction of the Huzinaga basis is used for fluorine. A set of d functions and a set of diffuse p functions are also added, bringing the total number of basis functions to 30. In the calculationson the C14Ti20system, the Cl-Ti distance is chosen to be 2.28 A from an independent geometry optimization on a similar molecule,' and the Cl-Ti-Cl angle is kept fixed at 120'. The only structural variable is the T i 4 separation, which covers the range from 1.65 to 1.95 A in increments of 0.1 A. The titanium basis set8consists of three 4s Gaussians contracted double { (2,1), two 4p Gaussians (double 0,and four 3d functions (double ( 3 ~ ) ) .The are electrons are replaced by an effective potential? Oxygen is treated with a basis set analogous to the fluorine basis! Chlorine is described by a minimum basis set for the valence space plus an effective potential for the core.9 Heisenberg Coupling Constants The energy gaps between spin states in molecules are related to the magnetic quantity J by the Heisenberg Hamiltonian? For pure spin states with an even number of electrons, this relationship is given by E(Smax)
The ligand spin polarization contribution to the coupling constant, JLSP, is
- E(S=O) = -Smax(Smax + 1)J
AE = E(S=l) - E(S=O) = -25
Therefore, the second-order ligand spin polarization energy difference between the high spin wave function and the broken symmetry wave function is E&P(Smax)
- EB:tSP = -SfnaxJLSP
(26)
The total Heisenberg coupling constant can be represented as the sum of contributions J = JF
+ fi; + &y+ JR
(27)
where JRcontains all the other unspecified contributions to the coupling constant. Therefore, the energy difference between the high spin state and the broken symmetry wave function for a given system can be used to calculate a coupling constant containing three important contributions: E(Smax)
- E B = -SmaxZ(JF + 6;+ JEF')
(28)
Applications of this theory have involved the use of Xa or local density functional (LDF) theory for calculating the wave functions. A problem with this approach is that throughout the derivation of relationship 28, the wave function for the high spin state was assumed to be a pure spin wave function, Le., an eigenfunction of spin. In the work published to date, the high spin state has been approximated by a UHF high spin wave function, which is definitely not an eigenfunction of spin.
(29)
The test cases investigated here involve one unpaired electron per center which can couple into a singlet or a triplet state. The following simple approximate relationship exists between the singlet-triplet energy splitting (AE) and J for the pure spin wave functions: (30)
The singlet-triplet energy splittings corresponding to magnetic interactions are quite small (1A1 5 1000 cm-l).l0 The choice of wave functions has been shown to have a profound effect on the sign and magnitude of J. The mechanism derived by Anderson," and later adapted by HTH,I2begins with the calculation of a RHF wave function for the high spin state. Configuration interaction, where the magnetic orbitals of the RHF high spin wave function are correlated in the perfect pairing approximation, is then used to obtain a wave function for the singlet state. J is determined from the energy difference of the two wave functions from eq 30. The contributions to J contained at this level of theory have been divided into a ferromagnetic piece (potential exchange) and an antiferromagnetic piece (superexchange). (Note: Several different terminologies have appeared in the literature to describe the same quantities. What is called potential exchange here has often been referred to as direct exchange, and superexchange in this context has also been called kinetic exchange.) Recent work has revealed that more extensive improvements on the RHF wave function are necessary to estimate J.2 Correlation involving the bridging ligand orbitals has been shown to have an impact on the magnetic coupling. One mechanism for this correlation is ligand spin polari~ation,2',~ which yields an antiferromagneticcontribution to the magnetic coupling. Other bridge effects also exist which can be either ferromagnetic or antiferr~magnetic.~,~ Another approach for calculating J involves a broken symmetry UHF wave function and a separate wave function describing the pure high spin statea3s4There are two important aspects of this approach. First, the broken symmetry UHF wave function is not an eigenfunction of spin, and spin projection must be used to obtain a relationship similar to (30) between the energy gap of the wave functions and the quantity J. By approximating the broken
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6261
Estimation of Magnetic Exchange Coupling Constants symmetry wave function as being made up of a weighted linear combination of spin states with S = 0 and S = S-, the following is ~ b t a i n e d : ~ . ~
E(Sma,)- E(broken symmetry UHF) = -(Sm,,)'J (31) Substituting S,, = 1, the result is E(E-1) - E(broken symmetry UHF) = -J
(32)
The second feature of the broken symmetry UHF approach is that it offers an approximation to a limited configuration interaction. It has been shown that the broken symmetry UHF wave function contains the potential exchange (PE), superexchange (SE), and ligand spin polarization (LSP) contributions found to be essential in estimating J. Although there are effects which are not included, J values calculated from two SCF wave functions should compare favorably to those obtained from a much larger CI treatment. Wave Functions For each molecule, the energies of wave functions at different levels of sophistication are used to calculate Jvalues. To obtain results following the formalism of Anderson" or HTH," an open shell RHF triplet wave function ( \ k m F ) is calculated. The singlet is described by a wave function (\ksRHF) which correlates the magnetic orbitals in the perfect pairing appr~ximation'~ (GVBPP( 1/2)). These wave functions are both eigenfunctions of the total spin operator with S ( \ k m F ) = 1 and S(\ksMF) = 0, SO (30) can be used to calculate JRHF. Included in this description are the potential exchange and superexchange contributions. The configuration interaction calculations carried out are different for each molecule. For H-He-H, there are only four electrons and eighteen basis functions, making full CI calculations practical for the singlet (\kscI)and the triplet (\kTcI). Two new wave functions are calculated to determine the CI basis, \kSGVB and \kTGVB, where the bridge orbitals are correlated in the perfect pairing approximation at the SCF stage. Since all possible configurations are used, all bridge effects are included in using (30) to obtain JcI. A subset of the full CI must be used for the [H-F-HI- anion due to computational limitations. The CI wave functions \kScI and \kTcI for the anion are based on GVB wave functions similar to the helium wave functions, where the bridge p orbitals are correlated. The configurationsincluded are limited to those arising from single and double excitations from the three possible occupations of the hydrogen GVB-PP orbitals (superexchange configurations) multiplied by single excitationsfrom the valence space. The valence space is defined here as the fluorine p orbitals and the hydrogen orbitals. This type of wave function is referred to as a polarization(3/2)*superexchange-CI(POL(3/2)*SE-CI), where 3 is the total order of the excitation relative to the three superexchangeconfigurations and 2 is the total number of electrons which are allowed into the virtuals in any one configuration. The coupling constant calculated from (2) for these wave functions (JcI)includes contributions coming from potential exchange, superexchange, and ligand spin polarization, as well as other bridge effects. For C14Ti20, the CI must be restricted further due to the increased number of orbitals (51) and active electrons (8). The basis for CI is also determined from a GVB wave function where the bridging oxygen orbitals are correlated in the perfect pairing approximation at the SCF level. However, due to computational limitations, the basis was reduced to 43 orbitals by omitting the d functions on oxygen which are not of the correct symmetry to interact with the occupied titanium orbitals. The 2s virtual orbital of oxygen character was also left out of the CI, along with the b-type virtual d orbitals on titanium. The oxygen s and chlorine orbitals are not included in any of the excitations, i.e., they are treated as core orbitals. The CI wave functions themselves are a slightly restricted version of the fluorine CI's. Instead of using the three superexchange configurations multiplied by all valence singles as a generating level for the singles and doubles excitations, the singles in the valence space are restricted to be within the GVB
pairs, This is called a restricted POL(3/2)*SE-CI. There are 25 627 spin eigenfunctions for the singlet and 28 951 spin eigenfunctions for the triplet. Even with these restrictions, JcIstill contains the contributions from potential exchange, superexchange, and bridge effects including ligand spin polarization. The above wave functions are eigenfunctions of spin, and the relationship defined by eq 30 is used to calculate J values. In contrast, calculating a J value via eq 3 1 uses the energy of a broken symmetry low spin wave function, where spin density of opposite sign is allowed to localize on each end of the molecule, and the energy of a high spin wave function obtained from a separate calculation. In previous work3 where eq 31 has been applied, the high spin state has been approximated by a UHF high spin wave function, which is not an eigenfunction of spin. However, the derivation of eq 3 1 assumes the high spin wave function to be an eigenfunction of spin? with S = 1 for our model compounds. In this section, the energy differences between the low spin broken symmetry wave function and two different high spin wave functions are used to calculate two different J values: J U H F , where the high spin state is approximated by a UHF high spin wave function, and JRU,where the high spin state is represented by the open shell RHF triplet wave function, which is an eigenfunction of spin. For H-He-H and [H-F-HI-, the UHF broken symmetry low spin (\kBUHF) and UHF high spin (*HUHF) wave functions are calculated. The molecule Ti2C&0is described by slightly modified UHF wave functions, where spin polarization of the chlorine orbitals is not allowed to occur during the calculation. This is accomplished by taking the orbitals of chlorine character obtained from the RHF triplet wave function and inserting them into the unrestricted high spin and broken symmetry low spin wave functions. Only the orbitals of titanium and oxygen character are allowed to change shape during the UHF calculations. The freezing of the chlorine orbitals is done to be consistent with the CI calculations, where the chlorine orbitals are also kept frozen at the R H F level. In this manner, the same orbital space is considered for both the CI and broken symmetry wave functions. In summary, four different coupling constants are calculated from the various wave functions for all the molecules using the appropriate expressions. The quantities and the corresponding wave functions used to determine them are JRHF JCI JUHF JRU
= -[E(*TRHF) - E(*SRHF)I /2
(33)
= -[E(*TcI) - E(*sc1)1/2
(34)
= -[E(\kHUHF) - E(*BUHF)I
(35)
= -[E(*TRHF) - E(*BUHF)I
(36)
Spin Contamination of the Broken Symmetry Wave Function
A simple relation~hip'~ exists between the expectation value for the total spin angular momentum ( ( S 2 ) )and the overlaps of the a and /3 orbitals (Sij)in the broken symmetry UHF wave function (S2)BuHF
=
(S2)exact
+ NU- CCISijI' i J
(37)
where = N , - NB= 0 for the low spin wave function and N, is the number of B electrons. (The number of a electrons N, is taken to be greater than or equalto NB.) The results of applying this relationship are tabulated for each of the molecules, and (S2)uHFis found to be close to 1.0. This is consistent with assuming \kBUHF to be an equal admixture of a pure singlet with (S2)s=o = 0 and a pure triplet with (Sz)s=l= 2. Magnetic Coupling Constants for H-He-H The energies of the wave functions at all the geometries are shown in Table I. Coupling constants are calculated using the appropriate relationships (eqs 33-36). The J values for H-He-H obtained from the calculated singlet-triplet gaps from the various wave functions are measured against those obtained from the full CI calculation (JcI) (Table 11, Figure 1). Each calculation produces similar results, but the combination of wave functions which gives a J closest to that of
Hart et al.
6268 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
TABLE I: Energies (au) of the Wave FUIIC~~OM for H2He R(H-He) RHF UHF GVB CI High Spin -3.769535 -3.770424 .. . . . ~ -3.815 136 -3.815 514 -3.828 183 -3.828412 -3.837 170 -3.837 305 -3.843312 -3.843 390 -3.847 526 -3.847482 -3.850296 -3.850 321 -3.852 182 -3.852 196 -3.854262 -3.854266 -3.855 149 -3.855 151 -3.855 520 -3.855 520 -3.855 741 -3.855 742
1.250 1.500 1.625 1.750 1.875 2.000 2.125 2.250 2.500 2.750 3.000 5.000
-3.779923 -3.827 358 -3.841 125 -3.850696 -3.857 300 -3.861 826 -3.864 909 -3.866 993 -3.869 321 -3.870335 -3.870 768 -3.871 040
-3.809217 -3.851 108 -3.863 025 -3.871 234 -3.876 862 -3.880 704 -3.883 3 13 -3.885 072 -3.887021 -3.887 853 -3.888 194 -3.888 369
-3.798 784 -3.831 880 -3.843 257 -3.851 689 -3.857 758 -3.862036 -3.865 004 -3.867 036 -3.869 330 -3.870 337 -3.870 768 -3.871 040
-3.831 008 -3.856 297 -3.865 444 -3.872 351 -3.877 379 -3.880945 -3.883 427 -3.885 127 -3.887 035 -3.887 857 -3.888 196 -3.888 369
Low Spin -3.779 185 -3.8 17 5 11 -3.829 340 -3.837 731 -3.843 584 -3.847615 -3.850361 -3.852214 -3.854 270 -3.855 151 -3.855 520 -3.855 742
-3.784 394 -3.8 18 8 16 -3.829 944 -3.838000 -3.843 699 -3.847 661 -3.850378 -3.852219 -3.854 269 -3.855 151 -3.855 520 -3.855 742
1.250 1.500 1.625 1.750 1.875 2.000 2.125 2.250 2.500 2.750 3.000 5.000
TABLE 11: JValues (em-’)for H2He R(H-He) RHF UHF 1.250 1.500 1.625 1.750 1.875 2.000 2.125 2.250 2.500 2.750 3.000 5.000
-1646 -404 -193 -9 1 -4 3 -20 -9 -4 -1
-1923 -439 -204 -94 -43 -19 -9 -4 -1
0
0 0 0
0 0
RU
CI
-2118 -521 -254 -123 -60 -29 -14 -7 -2 0 0 0
-2391 -569 -265 -123 -57 -26 -12 -6 -2 0 0 0
TABLE ILk Energies (au) for HgF R(H-F) RHF UHF High Spin 1.500 1.625 1.750 1.875 2.000 2.125 2.250 2.375 2.500 5.000
-100.428 236 -100.435 324 -100.439915 -100.442712 -100.444296 -100.445095 -100.445410 -100.445 441 -100.445 315 -100.442804
-100.431 435 -100.438 578 -100.442862 -100.445 146 -100.446 172 -100.446471 -100.446386 -100.446 119 -100.445 778 -100.442804
1.500 1.625 1.750 1.875 2.000 2.125 2.250 2.375 2.500 5.000
-100.445 928 -100.446051 -100.445788 -100.445692 -100.445705 -100.445 700 -100.445 626 -100.445482 -100.445 286 -100.442804
-100.439 113 -100.442784 -100.444846 -100.445915 -100.446358 -100.446420 -100.446 256 -100.445988 -100.445 668 -100.442804
GVB
CI
-100.460806 -100.469077 -100.474981 -100.479053 -100.481 784 -100.483 572 -100.484714 -100.485423 -100.485 843 -100.485 312
-100.589900 -100.595 588 -100.598 633 -100.599 819 -100.599 797 -100.599055 -100.597 964 -100.596794 -100.595 702 -100.590 530
-100.478 732 -100.478 429 -100.482 607 -100.484279 -100.485 737 -100.486888 -100.487 729 -100.488 307 -100.488680 -100.488 177
-100.618 941 -100.615736 -100.61 1603 -100.607 680 -100.604 343 -100.601 594 -100.599357 -100.597 565 -100.596 158 -100.590 8 16
Low Spin
TABLE Iv: J Values (cm-’) for [H2Fr R(H-F) RHF UHF 1.500 1.625 1.750 1.875 2.000 2.125 2.250 2.375 2.500 5.000
-1975 -1177 -644 -327 -155 -66 -24 -4 3 0
-1685 -923 -435 -169 -4 1 11 29 29 24 0
RU
CI
-2387 -1637 -1082 -703 -453 -291 -1 86 -120 -77 0
-3187 -221 1 -1423 -863 -499 -279 -153 -8 5 -50 -3 1
TABLE V: Energies (+3609 au) for ThCLO R(Ti-0) RHF GVB UHF High Spin 1.65 1.75 1.85 1.95
-0.1 14 266 -0.138851 -0.140230 -0.125555
-0.1 56 698 -0.183234 -0.186655 -0.174060
1.65 1.75 1.85 1.95
-0.114480 -0.138902 -0.140284 -0.125581
-0.156885 -0.183318 -0.186689 -0.174067
CI
-0.114 736 -0.139321 -0.140722 -0.126063
-0.250 662 -0.274945 -0.277063 -0.266845
-0.114865 -0.139370 -0.140723 -0.126044
-0,252206 -0.276284 -0.278295 -0.265660
Low Spin
E) 4-JUHF
-2000
fi
I--a-JCI
I
1007
f~
-2500 2.5
3
3.5
4
4.5
5
R (A)
P
Figure 1. J values for H-HeH as a function of R(H-He) distance.
full CI is the value obtained from the energy difference between the open shell RHF triplet and the UHF broken symmetry low The separation between JcIand J R U increases spin state (JRu). at shorter H-He distances, where the magnitude is near borderline between magnetic interactions and bonding @(triplet) - E(singlet) = 1000 cm-I). The J R H F and .IUHF values differ the most from the JcI results, but they still predict the interactions to be antiferromagnetic ( J < 0). At R(H-He) = 1.5 A, the magnitude of J R H F is -70% of Jc,.
Magnetic Coupling Constants for [H-F-Hr The energies for all the wave functions at each geometry are displayed in Table 111, and the J values are shown in Table IV. The differences in the J values for the [H-F-HI- molecule are ~ closely matches more pronounced (Figure 2). Again, J Rmost
-1100
-
1
5
3.5
3.75
0 4
-A
- J CI
2
,
4.5
4.75
5
0 4.25 R (A)
Figure 2. J values for H 2 Fas a function of R(H-F) distance.
JcI.J U H F deviates most from Jcr;it differs by a large number in magnitude, and there exists a region where it is of the wrong sign. Even though e H U H F is almost pure triplet, using the UHF wave function introduces spin polarization and lowers the energy of the high spin state with respect to the broken symmetry state. The other wave functions yield J values which are much smaller in magnitude than Jc, at all R.
J . Phys. Chem. 1992, 96, 6269-6278 TABLE VI: J Vdws (em-')for T i 2 Q 0 R(Ti-0) RHF UHF -23 -1 1
1.65 1.75 1.85 1.95
-28 -1 1
-6
0
-3
4
RU
CI
-131 -1 14 -108 -107
-169 -147 -135 -130
--f, - J U H F . . 0 . .J RU -6-JRHF-A-JCI
4
0
.-
-200-
I
1.65
1.70
1.75
1.80
1 1.85
1.90
1.95
R (4 Figure 3. J values for Ti2C140as a function of R(Ti-0) distance.
Magnetic Coupling Constants for TizCbO The results from the calculations for the transition metal system Ti2C140(Table VI, Figure 3) follow the trend suggested by the calculations on H-He-H and [H-F-HI-. The best agreement with Jcris again achieved by JRU. JRU covers the range from -107 to -131 cm-', while the CI results are between -130 and -169 cm-'.J R H F and JUHF are much smaller than JcI;they range from -3 to -23 cm-I and from +4 to -28 cm-', respectively. Conclusions We have calculated the energies of the singlet and triplet spin states of three model compounds using various wave functions. Coupling constants have been obtained using the appropriate energetic relationships. We have found, as derived by Noodleman and Davidson: that the J value obtained from the energy of a high spin restricted Hartree-Fock wave function and a low spin
6269
broken symmetry wave function, JRu,compares favorably to Jcr over a wide range of ion-bridge separations. If the energy for the high spin state is obtained from an unrestricted wave function, the m&itude of the resulting coupling constant, JmF,is too small; and it can be of the wrong sign. Superexchange, direct exchange, and ligand spin polarization have been shown to be included in the broken symmetry approach, if the high spin energy is obtained from a wave function which is an eigenfunction of spin. These contributions plus others are included in our CI calculations, but the computation time involved is much greater than with the S C F calculations. The broken symmetry approach offers an adequate approximation for these effects at a much lower cost. References and Notes (1) Murray, K. S. Coord. Chem. Reu. 1974. 12. 1. (2) (a) DcLoth, P.;Cassoux, P.; Daudey, J. P:; Malrieu, J. P. J . Am. Chem. Soc. 1981,103,4007. (b) De Loth, P.; Daudey, J. P.; Astheimer, H.; Walz, L.; Haase, W. J. J . Chem. Phys. 1985, 82, 5048. (c) Charlot, M. F.; Verdaguer, M.; Journaux, Y.; De Loth, P.;Daudey, J. P.Inorg. Chem. 1984, 23, 3802. (d) De Loth, P.;Karafiloglou, P.;Daudey, J. P.; Kahn, 0. J . Am. Chem. SOC.1988, 110,5676. (e) Astheimer, H.; Haase, W. J . Chem. Phys. 1986.85, 1427. (3) (a) Noodleman, L. J . Chem. Phys. 1981, 74, 5737. (b) Norman, J. G., Jr.; Ryan, P. B.; Noodleman, L. J. Am. Chem. Soc. 1980,102,4279. (c) Aizman, A.; Case, D. A. J. Am. Chem. Soc. 1982,104,3269. (d) Noodleman, L.; Baerends, E. J. J . Am. Chem. Soc. 1984,106,2316. (e) Noodleman, L.; Norman, J. G., Jr.; Osborne, J. H.; Aizman, A.; Case, D. A. J . Am. Chem. SOC.1985, 107, 3418. (4) Noodleman, L.; Davidson, E. R. Chem. Phys. 1986, 109, 131. (5) Huzinaga, L. J . Chem. Phys. 1965, 42, 1293. (6) Dunning, T. H.; Hay, P.J. In Methods ofBlectronic Structure Theory; Schaefer, H. F., Ed.; Plenum Press: New York, 1977; Vol. 4, Chapter 1. (7) Hart, J. R.; Rap#, A. K. Unpublished results. (8) Upton, T. H.; Rap#, A. K. J. Am. Chem. Soc. 1985, 107, 1206. (9) Rap#, A. K.; Smedley,T. A.; Goddard, W. A. J . Phys. Chem. 1981, 85, 1662. (10) Kahn, 0.; Briat, B. J . Chem. SOC.,Faraday Trans. 2 1976,72,268. (11) Anderson, P. W. Solid State Phys. 1963, 14, 99. (12) Hay, P. J.; Thibeault, J. C.; Hoffmann, R. J. Am. Chem. SOC.1975, 97, 4884. (13) Bobrowicz, F. W.; Goddard, W. A. In Methods of Electronic Structure Theory; Schaefer, H. F., Ed.; Plenum Press: New York, 1977; Vol. 4, Chapter 4. (14) Yamaguchi, K.; Toyoda, Y.; Fueno, T. Chem. Lett. 1986, 625.
Beryllium-Beryllium Bonding. 1. Energetics of Protonation and Hydrogenation of Be, and Its Ions Pablo J. Bruna, Gin0 A. Di Labio, and James S. Wright* Ottawa-Carleton Chemistry Institute, Carleton University, Ottawa, Ontario, Canada K l S 5B6 (Received: February 27, 1992) This paper is a theoretical study of beryllium-beryllium bonding, with emphasis on how to strengthen that bond. It deals with the structures and stabilities of several Be2H,Qspecies (with n = 1, 2 and charge q = -1, 0, +l). The ground states of Be2H, Be2H+,Be2H-, Be2H2,and Be2H2-are linear, whereas that of Be2H2+(like Be2H,) is bridged. The Be-Be bond in the linear compounds arises from sp, hybridization on Be, whereas in the bridged isomers it also has pr contribution (sp2 hybridization). Hydrogenation of the weakly bound Be2 (& = 4.63 bohr, we = 276 cm-',De = 0.1 eV) strengthens the BeBe bond significantly, as shown in the hydrides by an average &(BeBe) of 4.0 bohr for linear and of 3.8 bohr for bridged isomers. The frequencies w,(BeBe) lie in the range 650 f 50 cm-'. Hydrogenation to produce BezH leads to a D,(Be-Be) of 1.24 eV, while a second hydrogenation to produce HBeBeH strengthens it further to 3.20 eV. Both Be2H+and Be2H- also exhibit a strong D,(Be-Be) of ca. 3.30 eV. The process Be2H4(bridged) 2BeH2, however, is only 1.38 eV endothermic. The stability of these diberyllium hydrides mainly results from the withdrawal of antibonding charge upon Be2effected by the attached hydrogen atoms.
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1. Introduction He( Is2) and Be( ls22s2)are the first atoms from the periodic table with closed-shell ground states. One would therefore expect similarities in the bonding properties ofthe respective x2and X H molecules. Specifically, simple MO arguments indicate that each ground state should be repulsive. Such predictions are essentially 0022-3654/92/2096-6269$03.00/0
true for Hez(X'Z:,lu~l$,) and HeH (Xz2+,lu22u). For inStan% 0,and Re values of 0.95 meV (0.022 kcal/mol) and 5.60 bohr, respectively, have been reported for He2; studies on ground-state HeH show this species is also weakly bound.' Bez and BeH are considerably more stable than would be expected from their helium homologues. Bez(2$,2$,) has a De 0 1992 American Chemical Society
