J. Phys. Chem. 1989, 93, 2983-2997
2983
Electronic and Geometric Structures of the Metal Nitride Cations ScN', TiN', VN', and CrN' Kathryn L. Kunze and James F. Harrison* Department of Chemistry, Michigan State University, East Lansing, Michigan 48824- 1322 (Received: July 1 , 1988; In Final Form: October 13, 1988)
The electronic and geometric structures of various high-spin metal nitride cations [ScN', TiN', VN', and CrN'] with formal u,?r,r-triple, u,n-double, *,r-double, mingle, and mingle two-electron bonds have been studied by ab initio multiconfiguration self-consistent field and multireference configuration interaction techniques. In spite of different ground-state configurations of the bare metal cations and different calculated bond strengths (De),the species within each multibonded set exhibit remarkably constant bond lengths and intrinsic bond strengths (De*, De corrected for exchange energy loss upon bond formation and relative to the dN M' state). A Mulliken population analysis suggests the following: there is substantial ionic character in these bonds; regardless of the number of formal covalent bonds between N and a particular M', the gross M' to N charge transfer is constant; the equilibrium bond lengths all occur at the internuclear distance of maximum M' to N electron charge transfer; only at the equilibrium bond lengths is the M' to N electron charge transfer equally distributed over all formal bonds; the u bonds in the multibonded species have all du and no (0%) 4s character. The term symbol, calculated bond length (Rq, angstroms), vibrational frequency (we, cm-'), gross M' to N charge transfer, and the absolute bond energy relative to the ground-state fragments (De,kcal/mol) are as follows: Triply bonded species: CrN', 32-,1.596, 864,0.33, 49.5; VN', 2A, 1.574, 1005, 0.38, 87.1; TiN', ' Z ' , 1.585, 1039, 0.43, 97.8. u,r-Doubly bonded species: VN', 411, 1.738, 792, 0.37, 38.9; TiN', 3BI, 1.761, 822, 0.44, 55.8; ScN', 211, 1.802, 814, 0.49, 57.3. r,?r-Doubly bonded species: VN', 42-, 1.685, 812, 0.35, 34.5; TiN', 3A, 1.704, 863, 0.44, 54.3; ScN', 2Z+, 1.739, 871, 0.49, 63.1. u-Singly bonded species: ScN+, 4A, 2.088, 562, 0.54, 29.2. *-Singly bonded species: ScN', 4B,, 1.973, 682, 0.53, 22.2. Within the context of these calculations, we expect that the x,r-double bonds will persist in MNR' species, where R is a u-singly bonded species.
Introduction Diatomic molecules containing a single species bound to transition-metal cations are of considerable importance as models to further our understanding of the fundamental nature of chemical bonds involving transition elements. Gas-phase, bimolecular reactions of transition-metal cations with organic and inorganic molecules are being studied theoretically',2 as well as experimentally3 by using ion cyclotron resonance and guided ion beam techniques. The result is a growing thermochemical data base that is allowing the systemization of the observed chemistry ( I ) Publications in this series include: (a) Alvarado-Swaisgood, A. E.; Allison, J.; Harrison, J. F. J . Phys. Chem. 1985, 89, 2517. (b) AlvaradoSwaisgood, A. E.; Harrison, J. F. J . Phys. Chem. 1985,89, 5198. (c) Harrison, J. F. J. Phys. Chem. 1986, 90, 3313. (d) Mavridis, A,; AlvaradoSwaisgood, A. E.; Harrison, J. F. J . Phys. Chem. 1986, 90, 2548. (e) Alvarado-Swaisgood, A. E.; Harrison, J. F. J. Phys. Chem. 1988, 92, 2757. (f) Alvarado-Swaisgood, A. E.; Harrison, J. F. J . Phys. Chem. 1988, 92, 5896. (g) Alvarado-Swaisgood, A. E.; Harrison, J. F. THEOCHEM 1988,46, 155. (2) Recent publications of the theoretical groups besides our own currently active in this area include: (a) Schilling, J. B.; Beauchamp, J. L.; Goddard, W. A., I11 J . Am. Chem. SOC.1987, 109,4470. (b) Schilling, J. B.; Goddard, W. A., 111; Beauchamp, J. L. J . Am. Chem. SOC.1987, 109, 5573. (c) Schilling, J. B.; Beauchamp, J. L.; Goddard, W. A., 111 J . Am. Chem. SOC. 1987, 109,5565. (d) Carter, E. A,; Goddard, W. A., I11 J . Am. Chem. SOC. 1986, 108, 2180, 4746. (e) Pettersson, L. G. M.; Bauschlicher, Jr., C. W.; Langhoff, S. R.; Partridge, H. J . Chem. Phys. 1987,87,481. (f) Blomberg, J . Am. Chem. SOC.1987, 109, M. R. A.; Siegbahn, P. E. M.; Backvall, J.-E. 4450. (3) Recent publications of the experimental groups currently active in this area include: (a) Elkind, J. L.; Armentrout, P. B. J . Chem. Phys. 1987,86, 1868. (b) Sunderlin, L.; Aristov, N.; Armentrout, P. B. J. Am. Chem. SOC. 1987, 109, 78. (c) Aristov, N.; Armentrout, P. B. J . Phys. Chem. 1987, 91, 6178. (d) Reents, W. D.; Strobel, F.; Freas, R. B.; Wronka, J.; Ridge, D. P. J . Phys. Chem. 1985,89, 5666. (e) Hettich, R. L.; Freiser, B. S. J . Am. Chem. SOC.1987, 109, 3543. (f) Radecki, B. D.; Allison, J. Organometallics 1986, 5, 41 1. (g) McElvany, S.W . ;Allison, J. Organometallics 1986, 5, 1219. (h) Hanratty, M. A.; Beauchamp, J. L.;Illies, A. J.; van Koppen, P.; Bowers, M. T. J. Am. Chem. SOC.1988, 110, 1. (i) Schilling, J. L.; Beauchamp, J. L. J. Am. Chem. SOC.1988, 110, 15. (j)Schilling, J. L.; Beauchamp, J. L. Organometallics 1988, 7, 194. (k) Tolbert, M. A.; Mandich, M. L.; Halle, L. F.; Beauchamp, J. L. J . Am. Chem. SOC.1986, 108, 5675. (I) Kang, H.; Beauchamp, J. L. J . Am. Chem. SOC. 1986, 108, 5663. (m) Kang, H.; Beauchamp, J. L. J. Am. Chem. SOC.1986, 108, 7502. (n) Lebrilla, C. B.; Schulze, C.; Schwarz, H. J . Am. Chem. SOC.1987,109, 98. (0)Lebrilla, C. B.; Drewello, T.; Schwarz, H. Inr. J. Mass Spectrom. Ion Proc 1987, 79, 287. (p) Schulze, C.; Schwarz, H. J. Am. Chem. SOC.1988, 110, 67. (p) Stepnowski, R.; Allison, J. Organomefallics, in press.
0022-3654/89/2093-2983$01.50/0
and prediction of possible new chemistries with immediate impact in organometallic chemistry, surface chemistry, catalysis, and atmospheric chemistry. Theoretical studies in the field have lagged the experiments because the extent of electron correlation needed for even qualitatively correct results is significantly raised compared to calculations on molecules containing only main-group elements. Recent results' suggest that theoretical studies can provide usefully accurate descriptions of these interesting systems when they take into account some of the structural correlation of the transitionmetal-main-group bonds and all of the various valence spin couplings associated with those bonds and the valence spectator electrons consistent with the desired spin state. The purpose of this paper and others in this series' is to use all-electron ab initio electronic structure theory to provide insights into the electronic and geometric structures of representative transition-metal-containing cations. Now that studies of transition-metal cations with hydrogen and hydrocarbons, such as activation of hydrocarbons, have become numerous, la b d e 2a-c,3a-k experimentalists are extending their studies to molecules containing other main-group elements,31including n i t r ~ g e n . ~As " ~a step toward the development of a qualitative understanding of the nature of the chemical bonds between M+- and N-containing compounds, we investigated the bonding of M+ to N in a sequence of 11 species with formal two-electron single, double, and triple bonds between a single ground-state N and a high-spin s'dN-' or dN early-transition-metal +1 cation, Sc', Ti', V', or Cr'. We anticipate that this will result in our understanding the nature of the transition-metal cation-N bond sufficiently well so as to permit a qualitatively correct prediction of the geometry and bonding in more complex systems containing a transition-metal cation bonded to N. To our knowledge there are no experimental results on MN+ species. However, theoretical results predictIcv4 triply bonded FeN' to have a bond energy of only 22.8 kcal/mol with a bond length of 1.529 8, but triply bonded CrN' to have a larger bond energy of 48.5 kcal/mol and a bond length of 1.602 8,. The symmetries, bond strengths, bond lengths, spectroscopic properties, and bonding description of various MN' species should be useful in helping to predict possible chemistries, and com-
-
9
3
(4) Siegbahn, P. E.; Blomberg, M. R. Chem. Phys. 1984, 87, 189.
0 1989 American Chemical Society
2984
The Journal of Physical Chemistry, Vol. 93, No. 8. I989
parisons should he useful in helping to explain differences in the reactivities of the metal ions with N-containing species. After a preliminary introduction to the molecular species studied, each molecule is described, including the multiconfiguration SCF (MCSCF) wave function construction, the potential curves, and the atomic orbital populations along the potential curves. Then general results are presented, and comparisons of results within the series are explored. General computational details, including the basis sets, molecular codes, methods, and reliability, are described in the Appendix.
PreIim inar ies Eleven early-transition-metal nitride (+I) cations that result from the interaction of the high-spin stdN-l or dN states of Sc+ (IDor IF), Ti* (4F or 'F), V+ (IDor SF), and Cr+ (6S or 6D) with ground-state (high spin) N (2.s22p')(%) were studied. While the ground states6 of Sc+ and Ti+ arise from the configuration (AI core)3d"'4s1, those of V+ and Cr+ arise from the configuration (AI core)3dN. These calculations were carried out in C, symmetry with the molecular axis, the I axis, as the a i symmetry axis and with x and y as bl and bz, respectively. The M+valence orbital designation and symmetries are as follows: s is 4s (al), du is 3d2 (al), dux is 3d, (bJ, du, is 3dr (b?). 6+ is 3 d w (al), and 8- is 3d,, (ad. N valena orbital designation and symmetries are as follows: s is 2s (al), pu is 2p. (al), px is 2px (bl), and p, is 2pY( W . There are interesting orbital occupation and spin-coupling options available for the interaction of electrons on M+ with those on N. For example, Cr+ with an electron in each 3d orbital results in the 9- state of CrN+, which corresponds to ihe maximum number of bonds, formally three, with the two spectator 3d8 electrons on Cr+ triplet coupled 0,n.n Trtply-Bonded CrNc 32-
Y
the highest spin multiplicities possible consistent with the number of such bonds. Although we start with the lowest energy asymptote, as the two fragments approach one another, several electronic configurations of the fragments, including those that contribute to ionic MN* configurations, such as M2+N; can contribute to the variationally optimized MN+ wave function. For example, for the triply bonded 32-molecular state of CrN+, the molecular configuration along the adiabatic potential curve evolves to that appropriate to the molecular environment with possible contributions from components of the 3dS6S and 3d44si 6DCr+ (47-3du mixing), along with possible 2s-2pu hybridization of the N and charge transfer. For a given bond type, the structures for the different metal ions differ in the number of spectator electrons in the 3dd orbitals. For example, the triply bonded zA state of VN+ can be thought to arise by removal of the 3d6+ spectator electron from the triply bonded CrN+ state:
V+
N Singly and doubly bonded MN+ species can he thought to arise by breaking one or two bonds of the more highly bonded MN+ species. For example, u,r-doubly bonded VN+ species result by breaking the rxbond of the triply bonded VN' 2A state. This can be accomplished by simply unpairing the spins of the 3dn; and 2px bonding electrons, making them parallel spin spectator electrons that couple with the electron in the V* 3d6- to form a quartet state, 'A. Since this state has the same V+ orbital occupation as the 2A state but is higher in energy, we choose not to study it. A different V+ orbital occupation results if after unpairing the spins we also move the 3du, electron on V+ to the 3d6+ orbital, which is spatially out of the way of the bonds. We choose to study the resulting 'II state:
N
CI+
Dovh,, B m d d
(I."
The orbitals pictured are as follows: Cr+ 3dr, and N 2px orbitals are shaded grey: Cr+ 3dry and N 2py orbitals are unshaded; Cr* 3do and N 2pu orbitals are shaded black. The 6, and 6. symbols represent those singly occupied Cr+ orbitals. The '2- state has the same Cr+ orbital occupation as the state but corresponds to the formally o-singly bonded CrN+ where the four Cr+ and two N speclator electrons have o( spin. We assume that with fewer bonds the molecule bas a higher energy and do not consider it further:
c+
Kunze and Harrison
N
Vi
N
Another example, the 'r state of CrN+ has the same orbital occupancy as the %state and comsponds to a formal triple bond, but has the MB electrons singlet coupled and is, therefore, of higher energy also. It does not correlate with high-spin Cr+ and is not studied. Thus, for a particular M* orbital occupation the molecular species we chose to study has the largest number of two-electron chemical bonds (singlet-coupled pairs) possible and ~
~~~
( 5 ) Moore. C. E.Alomic Ewrgy his:Nai. Stand. Ref. Data Scr., Nat Bur. Stand.. Cir. 3% Washington D.C., 1911; Vol. I and 11. (6) Mullikn. R. S. J. Chem. Phyr. 19% 23. 1833. 1841.2338.2343. For a critique see. Noell. J. 0 lnorg. Chrm. 1982. 21. I I ,
Vt
N
Another example, the r s d o u b l y bonded VN* species results by breaking the u bond of the triply bonded VN+ 2A state. By unpairing the spins, the 3du and 2pu formerly bonding electrons become spectator electrons that couple with the electron in the V* 3d6. to form a quartet state, 'A, with same V* orbital m u pation as the 2A state. The ' t s t a t e results if we then move the 3du spectator electron on V+ to the 3d6+ orbital. Of the two possibilities, we choose to study the '2- state:
N
Cri
""*
1r
"0"hb hdd wIJ-
1"
V+
N
Vi
N
The spectroscopic term symbols and irreducible representations in C, symmetry for the MN' species studied are collected in Table I. Since real functions are used, in some instances the molecular state of the MN+ has a mixed (MI = 1.3) angular momentum
Metal Nitride Cations
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 2985
TABLE I: Metal Nitride +1 Cation Bond Lengths (angstroms) and Vibrational Frequencies (ue,cm-I)
bond length IRR
MCSCF +1+2*
MN+
state
C,,
MCSCF
CrN+
'Z-
VN+ TiN+
'A
1.574 1.563 1.575
1.596 1.577
P
'A2 2A2 'AI
VN+ TiN+ ScN+
411
4B2
1.745 1.763
MCSCF +1+2
vib freq MCSCF (N2S)+1+2
MCSCF +1+2*
MCSCF
MCSCF +1+2
MCSCF (N2S)+1+2
1005 1039
1045
MN+ Triple Bonds I
924 1043 1075
864 996
M N + a,*-Double Bonds 1.738 1.761 1.760 1.803 1.804
700 797 796
782 821
792 822 814
818 811
M N + *,*-Double Bonds 1.685 1.704 1.699 1.740 1.738
672 85 1 872
825 864
812 862 87 1
869 87 1
1.574 1.585
1.586
211
2B2
1.756 1.763 1.800
VN+ TiN+ SCN'
48-
3A '2'
4AI 'A2 'Ai
1.725 1.721 1.752
1.686 1.705
ScN+
'A
4AI
2.133
2.095
M N + d i n g l e Bonds 2.088 2.101
487
558
562
534
4B1
1.976
1.975
M N + *-Single Bonds 1.973 1.979
690
686
683
674
'Bl
ScN+
and only symmetry under C2, is designated.
The Molecules For each molecular species we discuss the MCSCF wave function construction and then present the MCSCF and MCSCF+l+2 potential curves. The character of the MCSCF potential curve can be understood by following the atomic occupations and the gross M+ to N charge transfer obtained from the valence natural orbitals (NOS) of the MCSCF wave function by Mulliken population analysis6 as a function of Cr-N internuclear separation. While the limitations of the Mulliken population analysis6preclude detailed arguments based on it, we are confident that the qualitative changes in the populations along the potential curve are real. The changes in the N O occupation numbers as a function of internuclear separation foreshadow the changes in the populations and the potential energy and provide additional insights into the distance dependence of the electron atomic occupations. CrN'. A. MCSCF Wave Function Construction. u,a,x- Triply Bonded CrN'. The CrN+ high-spin molecular state that will admit three formal two-electron bonds between unpaired electrons on Cr' in the (3ds) 6S or (3d44s1)6D state and unpaired electrons on N in the (2s22p3)4S state has 32-symmetry and is calculated , symmetry. The structural correlation of the bonds as 3A2under C is provided as in generalized valence bond (GVB)7 manner. The asymptotic form of the GVB representation of the wave function is as follows:
-
CrN': 3.X- (3A2) [ 4 M ) d N ) + 4 N ) 4 M ) 1 [xy(M)xy(N) + xy(N)xy(M)I X [*x(M)ax(N) + rx(N)xx(M)l 3d6+ 3d6where M = Cr; the three spin pairs are (1) o(M) and u(N), (2) a,(M) and ay(N), and (3) xx(M) and x,(N), and the 3d6 orbitals are triplet coupled. At large separation o(N) is the 2pa orbital and u(M) is the 3do in the 6S ground state; ay(M) is the 3duy orbital and uy(N) is the 2py orbital; o,(M) is the 3da, orbital and x,(N) is the 2p, orbital. The 3d6+ and 3d6- orbitals are the Cr+ 3d,z+ and 3d,, respectively. These eight orbitals constitute the valence space. In this and in subsequent discussions we will suppress the expression of the 18 electrons in the Ar core on Cr+ and the 4 electrons in the Is and 2s orbitals on N. These are, of course, included in the calculation and are variationally optimized. This GVB wave function does not separate to the pure (7) (a) Gcddard, W. A., 111; Dunning, T. H.; Hunt, W. J.; Hay, P. J. Acc. Chem. Res. 1973, 6, 368. (b) Hay, P. J.; Hunt, W. J.; Goddard, W. A,, I11 J. Am. Chem. Soc. 1972, 94, 8293. (c) Bobrowicz, F. W.; Gddard, W. A., 111 In Modern Theorelicd Chemisfry;Schaefer, H. F., 111, Ed.; Plenum Press: New York, 1977; Vol. 3, Chapter 4.
P
020-
I
E
MCSCF
-
MCSCF+ 1 c2'
.-c 40a . n 60 -
0.0
20
4.0
R(Cr-N)
60
8.0
10 0
(au)
Figure 1. Potential energy curves for the triply bonded state of CrN+ as a function of internuclear separation for the M C S C F and MCSCF+1+2* calculations. The energy units are millihartrees; 1
mhartree equals 0.6257 kcal/mol. spin states of the fragments and leaves the perfectly paired spins of the bonds undercorrelated compared to the parallel spins in the 3d6 orbitals of Cr+. A multiconfiguration self-consistent field (MCSCF) function of this form provides the structural correlation of the bonds in the GVB fashion7 and includes all possible spin couplings among the eight valence electrons consistent with the desired spin state, including that of the spectator electrons in the singly occupied 3d6 orbitals with the bonding electrons. Thus, the atomic orbitals in the active space comprise all eight valence orbitals. The MCSCF wave function consists of 126 configuration state functions (CSFs) and dissociates properly to the sum of the single self-consistent field (SCF) configuration functions for the atoms [Cr+('%) and N(4S)]. B. Potential Curves. The MCSCF function is constructed and solved variationally as a function of the Cr-N separation. The MCSCF potential curve relative to the sum of the ground-state fragment S C F energies and the MCSCF+1+2* potential curve relative to the sum of the corresponding ground-state fragment energies are shown in Figure 1. MCSCF+1+2* denotes a 126-configuration MCSCF reference space singles and doubles configuration-interaction calculation which is described in the Appendix. The dissociation energy is 34.3 and 49.5 kcal/mol at the MCSCF and MCSCF+ 1+2* levels, respectively. As noted in the Appendix, the calculations are size-consistent. C. Analysis of the MCSCF Wave Function along the Potential Curve. The electron populations in the o and one set of the identical a atomic orbitals and the gross Cr+ to N charge transfer are shown in Figure 2. The changes in the N O occupation numbers as a function of internuclear separation for the u orbitals and one set of the identical x orbitals are shown in Figure 3. As the N approaches Cr' from infinite separation until a separation of 5.00 au, the N O occupation numbers are only very slightly diverging from 1.0 and the potential curve has a slight slope,
2986
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 1.4-
I
R(Cr-N)
(au)
Figure 2. Electron population of selected atomic orbitals of u and R symmetry from the valence natural orbitals of the 126 CSF MCSCF
wave function and gross Cr' to N charge transfer of triply bonded 3ZCrN' as a function of internuclear separation. R, designates the calculated bond length. Note: only one of the two identical R'S is pictured. 2.0
,
1
Kunze and Harrison all three bonds, reflecting the spherical nature of both Cr' and N. Also, the maximum (0.33 electron) in the gross Cr+ to N valence charge transfer as a function of internuclear distance occurs at the equilibriuni bond distance. That the population of the Cr' 4s is negligible at all internuclear distances indicates that the Cr+ 6D (d4s1)configuration does not mix with the 6S (d5) configuration. That the population of the N 2s remains 2.0 indicates that the N does not hybridize at any internuclear distance. Thus the character of the u-bonding orbitals appropriate to the molecular environment turns out to be pure Cr+ 3du and pure N 2pu. EN+. A . MCSCF Wave Function Construction. u,rr,a-Triply Bonded TiN+. The wave function for triply bonded TiN' (IA, under C2, symmetry) is constructed in a manner similar to that for triply bonded CrNf 32-but without both spectator d6 electrons. At large separation u(M) is 4s. Under C, symmetry, ~ 'a, mixture of the 4F and the Ti' orbital occupation, 4 ~ l d a , ~ d r r is 4P 3d24s1lowest states of Ti', (4/5)'/'I4F) - (1/5)1/214P).The MCSCF function of this form separates to the S C F products [Ti+((4/5)1/214F)- (1/5)1/214P))+ N(4S)]. u,a-Doubly Bonded T i w . The high-spin TiN' molecule with one u and one a spin pair between N in the 4S state and Ti' in either the (3d3) 4F excited state or else having the (3d24sl) 4 ~ l d a ~ ~ d 6occupation, -l which under C2, symmetry is also a mixture of the 4F and 4P states of Ti', (3/5)'/214F) + (2/5)1/214P), has 3B1 symmetry in Czo. The asymptotic form of the GVB representation of the wave function is as follows: TiN? (3B,)
-
[a(M)u(N) + u(N)u(M)] X [a,(M)a,(N) + *,(N)*,(M)I
3d6- 2Py
where M = Ti and the extra electrons in the N 2py and Ti+ 3d6orbitals are coupled into a triplet. An MCSCF function of this form that permits all spin couplings among the six valence electrons separates to the S C F products [Ti'((3/5)1/2)4F) (2/5)'/'I4P)) N(4S)] where u(M) is 4s. a,a-Doubly Bonded T i p . The high-spin TiN' molecule with two a spin pairs between Ti' in the (3d3) 4F excited state and 4S N is a 3A state (3A2in C2,). The asymptotic form of the GVB representation of the wave function is as follows:
+
+
00
20
4'0
R(M-N)
60
80
-
100
Figure 3. Occupation numbers of the natural orbital pairs corresponding to formal bonds of u and R symmetry for the 126 CSF MCSCF wave function of triply bonded 3X- CrN+ and for the 37 CSF MCSCF wave function of triply bonded 'X' TiN' as a function of internuclear separation. R, designates the calculated bond length. Note: only one of the two identical T'S is pictured for each.
indicating increasing energy stabilization due to electrostatic attraction but no covalent or ionic bonding interaction. At 5.00 au the N O occupation numbers begin to diverge for both the u and the a systems, but the singly occupied a-bonding orbitals remain pure 2pu and pure Cr+ 3du and the N 2pu orbitals and the Cr+ 3da orbitals remain singly occupied. The N O occupation numbers diverge most rapidly for the u system between 4.50 and 3.75 au, and when that of the a-bonding partner NO is 1.4 at 4.25 au the potential energy begins to drop with the onset of Cr' to N electron transfer in the u system. By 4.00 au the energy is plunging due to the formation of the u bond. For the a system the N O occupation numbers diverge most rapidly between 4.00 and 3.25 au, and when that of the a bonding N O is 1.4 at 3.75 au a gradual Cr' to N electron transfer in the a systems begins. The potential energy continues to drop due to u- and rr-bonding interaction until a Cr-N separation of 3.25 au, where the electron population in the N orbitals has increased to a maximum of 1.15 electrons in the 2pu, while that in the Cr' 3du orbital has dropped to 0.8. Then as the molecule continues to approach the equilibrium Cr-N separation of 2.924 au, the charge transfer in the u system decreases, whereas Cr' to N charge transfer in each of the a bonds has continued to increase. At equilibrium the charge transfer is 0.1 1 in the u bond and 0.10 in each of the a bonds. Thus, only at the equilibrium is the charge transfer equally distributed over
-
-
[ay(M)a,(N) + ay(N)ay(M)] X [a,(M)a,(N) + a,(Nhx(M)I 3d6- 2PU where M = Ti and the extra electrons in the N 2pu and Ti+ 3d6orbitals are coupled into a triplet. The MCSCF function of this form permits all spin couplings among the six valence electrons and separates to the S C F products [Tif(3d3,"F) + N(4S)], since the Ti+ (3d24s1)4F ground state does not contribute. B. Potential Curves. For the triply bonded and the u,adoubly bonded 3B, species the MCSCF potential curves relative to the sum of the Ti'((4/5)1/214F) - (1/5)1/214P))and N(4S) fragments and the MCSCF+1+2 potential curve relative to the energy of the TiN' supermolecule asymptote at R = 20.0 au are shown in Figure 4. For the r,a-doubly bonded 3A species the MCSCF and the MCSCF+1+2 potential curves relative to the ground-state fragments are shown in Figure 5. Of the TiN+ species studied, the one of lowest energy is predicted to be the triply bonded species (l2') with a dissociation energy of 97.8 kcal/mol at the MCSCF+1+2 level. The u,r-doubly bonded (3B,) and a,a-doubly bonded (3A) species have very similar dissociation energies of 55.8 and 54.8 kcal/mol, respectively. Note, however, that the n,a-doubly bonded TiN' separates to the excited-state fragments and is bound by 60.0 kcal/mol with respect to these excited-state asymptotic fragments. C. Analyses of the MCSCF Wave Functions along the Potential Curves. u,a,a- Triply Bonded TiN'. The Ti-N internuclear distance dependencies of the electron populations in the u and one set of the identical T atomic orbitals and the gross Ti+ to N charge transfer obtained from the MCSCF NOS in triply bonded TiN+ are shown in Figure 6. The distance of the MCSCF N O occupation numbers for the u orbitals and one set of the identical a orbitals for TiN' are shown in Figure 3, where they can be compared to that for CrN' 'Z-. For both plots we see that TiN': 3A(3A2)
(OU)
Metal Nitride Cations
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 2987
01
1
40 60-
I
4
E
.-c a
0
80,
.
04
100-
02
Charge+\
TI+ d
"7
I
'40L 00 00
1600.0
2.0
6.0
4.0
R(Ti-N)
(au)
Figure 4. Potential energy curves for the triply bonded I P state and the u,r-doubly bonded 3Bl state of TiN+ as a function of internuclear sep-
aration for the MCSCF and MCSCF+l+2 calculations.
w 3 AMCSCF
"1 801
i
,
,
;i, ,
?N+,
,
,
100
0.0
2.0
4.0
R(Ti-N)
10.0
8.0
6.0
(au)
Figure 5. Potential energy curves for the ?r,?r-doublybonded 3A state of
TiN+ as a function of internuclear separation for the MCSCF and MCSCF+1+2 calculations. 1.4
0.0
I
'
210
'
).,
410 ' 6jO
R(Ti-N)
'
8;O
'
10.0
(au)
Figure 6. Electron population of selected atomic orbitals of u and A symmetry from the valence natural orbitals of the 37 CSF MCSCF wave function and gross Ti+ to N charge transfer of triply bonded IZ+TiN' as a function of internuclear separation. R , designates the calculated bond length. Note: only one of the two identical A'S is pictured.
there is.a difference compared to CrN+ since the Ti+ asymptote has the 4s occupied rather than the 3da. However, the Ti-N bond is similar to that of CrN+ and is solely between the unhybridized N and the (3d') 4 F excited state of Ti+. At all internuclear distances the N 2s remains doubly occupied. The N O occupation numbers in the a system have significantly diverged when the Ti-N separation is decreased to 5.00 au, but the potential curve has only a slight slope and the singly occupied orbitals remain singly occupied. When the occupation number of the a bonding N O is
20
40
R(Ti-N)
10.0
8.0
Re
60
80
100
(au)
Figure 7. Electron population of selected atomic orbitals.of u and A symmetry from the valence natural orbitals of the 25 CSF MCSCF wave function and gross Ti+ to N charge transfer of u,r-doubly bonded 3BI TiN+ as a function of internuclear separation. R, designates the cal-
culated bond length.
-
1.4 at 5.00 au (a larger M-N separation than for CrN+), Ti+ 4s to N 2pa charge transfer begins, but the slope of the potential curve does not change. It is not until 4.50 au that the potential energy begins to drop as the Ti+ 3do orbital starts to become populated with continuing depletion of the 4s. The energy of ionic bonding via charge transfer supplies the 3d24s1to 3d3 promotion energy. Alternatively, one could think that the N 2pa could be the conduit for the shift of electron population on Ti+ from the 4s to the 3da, rather than direct transfer. The N O occupation numbers diverge most rapidly for the a system between 4.50 and 4.00 au and that for the n system between 4.50 and 3.75 au. When the occupation number of the n bonding N O is 1.4 at 4.25 au, an initial reverse (N to Ti') charge-transfer blip appears. Electron transfer in the a system from Ti+ to N results in a maximum in the electron population in the N 2pa orbital of 1.36 at a Ti-N separation of 4.00 au. The plot of the a-bonding N O occupation numbers shows a jog, which indicates a change in character of the a system. At this distance the potential energy is plummeting concomitant with a dramatic change in the populations of the Ti+ orbitals of a symmetry as the 3da orbital takes over and the 4s is depleted. Then as the molecule continues to approach equilibrium, the electron population in the N 2pa orbital decreases and that in the 3da increases, while that in the Ti+ 4s orbital drops to zero. At 3.75 au a gradual transfer of electrons in the n systems from Ti+ to N begins, and the potential energy continues to drop due to both a- and n-bonding interaction. At the equilibrium bond length of 2.977 au, the Ti+ contribution to the a bond is all d a with no 4s or 4p, while the contribution of the N is all 2pa with no 2s (no hybridization of either Ti+ or N), despite the 4s'3d2 ground state for Ti+. Again, only at the equilibrium bond distance is the Ti+ to N charge transfer equally distributed over all three bonds, being 0.13 electron in the a bond and 0.14 electron in each of the R bonds. The total Ti+ to N charge transfer has a maximum (0.43 electron) at the equilibrium bond distance. a,n-Doubly Bonded T i p . The Ti-N internuclear distance dependencies of the electron populations in the a and R atomic orbitals and the gross charge transfer obtained from the MCSCF NOS in a,n doubly bonded TiN+ are shown in Figure 7. The distance dependence of the MCSCF N O occupation numbers for the a orbitals and the n orbitals for TiN+ are shown in Figure 8 and are similar to that for the triply bonded TiN+ except that divergence is greater after the Ti-N separation is decreased to 4.00 au, reaching the same numbers at the longer equilibrium bond lengths. The features in the population plots are also remarkably similar to that for the triply bonded TiN+, the differences being greater Ti+ to N charge transfer in the u and nx systems and the ny back-donation. At the equilibrium bond length of 3.332 au the Ti+ to N charge transfer in the a system (0.26 electron) is the same as that in the A, system (0.25 electron), but this condition is reached at longer bond lengths than for the triply bonded species. N
2988
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
Kunze and Harrison
C
._0 Y
1.5-
0.
3 U
u 0
-
.-c0
1.0-
e 0 -
?
;;
0.5-
z
- 0 . 0 0'0
20
4'0
R(M-N)
60
80
160
0.0
2.0
4.0
R(M-N)
(OU)
Figure 8. Occupation numbers of the natural orbital pairs corresponding to formal bonds of u and T symmetry for the 25 CSF MCSCF wave function of u,r-doubly bonded )B, TiN' and for the 17 CSF MCSCF
wave function of u,R-doubly bonded 211 ScN' as a function of internuclear separation R, designate the calculated bond lengths.
6.0
8.0
1
'LO
(au)
Figure 10. Occupation numbers of the natural orbital pairs corresponding to formal bonds of R symmetry for the 25 CSF MCSCF wave function of r,?r-doublybonded 'A TiN+ and for the 17 CSF MCSCF wave function of r,r-doubly bonded 2Zt ScN' as a function of internuclear separation. R, designate the calculated bond lengths. Note: only one of the two identical R'S is pictured for each.
1 4
I
I
',,
Ti' to N charge transfer reaches its maximum (0.44 electron) at the equilibrium bond distance. Again, this is the same magnitude as for the triply bonded TiN'. VN'. A . MCSCF Wave Function Construction. u,x,x-Triply Bonded Vn". The wave function for triply bonded VN' zA (*A2 under Czusymmetry) is analogously constructed from that for CrN' 32-by removing one spectator d6 electron. Since the V' 5D 3d4 and 5F 3d34s' S C F energies are not in the experimental order, the MCSCF function separates to the [V+(SF) N(4S)] SCF products where u(M) is 4s. In addition to the adiabatic curve we also constructed a "SD(3d4) character conserving" MCSCF solution that separates to the [V+(sD) + N(4S)] asymptote where u(M) is 3du. Note that the two calculations are the same below 4.25 au, and at 4.25 au they give the same energies with different wave functions. Since the SCF+1+2* and SCF+1+2 V' energies are in the correct order, the 4S N approaches V' in the 5D state at the CI levels. u,a-Doubly Bonded V W . The wave functions for u,a doubly bonded VN' 4 J l ;4B2 in Czu)is similarly constructed as the u,a,,-bonded analogue of TiN' 3B1with a spectator electron added in the second 3d6 orbital. The spectator electrons in the N 2p, and the two V' 3d6 orbitals are coupled into a quartet. The MCSCF function separates to the S C F products [V+(5F) N(4S)] where u(M) is 4s. a,a-Doubly Bonded VN'. The wave functions for x,a-doubly bonded VN' 42-(4Azin Czu)is similarly constructed from that of TiN' 3A by addition of another spectator 3d6 electron. The spectator electrons in the N 2pu and the two V+ 3d6 orbitals are coupled into a quartet. The MCSCF function separates to the S C F products [V+(5D) + N(4S)] since the V+ (3d34s1) SFstate does not contribute. B. Potential Curves. The potential curves for the three VN' species are shown in Figure 11. Since the d6, orbitals are of the same symmetry as the du, s, and pu orbitals, parking a spectator electron in the 3d6, orbital for the 411 and 48-doubly bonded states has resulted in contributions from contaminating configurations at long bond lengths and only the uncontaminated part of the potential curve is presented. The full adiabatic and "5D character conserving" MCSCF potential curves for the *A state and the equilibrium regions of the MCSCF potential curves for the 411 and 42- states are displayed relative to the sum of the S C F ground-state V'(5F) and N(4S) fragment energies. The corresponding MCSCF+ 1+2 potential curves are displayed relative to the SCF+1+2 ground-state V+(5D) and N(4S) fragments. Of the VN+ species studied, the one of lowest energy is predicted to be the triply bonded species (2A) with a dissociation energy of 87.1 kcal/mol at the MCSCF+1+2 level. The u,a-doubly bonded
+
R(Ti-N)
(ou)
Figure 9. Electron population of selected atomic orbitals of u and R symmetry from the valence natural orbitals of the 25 CSF MCSCF wave function and gross Ti+ to N charge transfer of R,r-doubly bonded 3A TiN* as a function of internuclear separation. R, designates the cal-
culated bond length. The gross Ti' to N charge transfer is maximum (0.44 electron) at equilibrium, the same magnitude as for the triply bonded TiN' species. At equilibrium the u bond has no Ti' 4s or 4p. In addition to the formal bonds, there is a small delocalization (back bonding) (0.08 electron) of the N 2py onto the Ti' 3dpy, which reduces the total charge transfer in the a system, making the u system the more polar. a,x-Doubly Bonded T i p . The Ti-N internuclear distance dependencies of the electron populations in the u and A atomic orbitals and the gross charge transfer obtained from the MCSCF ~ bonded TiN+ are shown in Figure 9. The NOS in a , doubly distance dependencies of the MCSCF N O occupation numbers for the a orbitals for TiN' are shown in Figure 10 and are similar to that for the a orbitals in triply bonded TiN+ except that divergence is greater at longer Ti-N separation, reaching the same numbers at longer bond lengths. When the Ti-N separation is less than 5.00 au and the bonding NO occupation numbers are 1.4, the potential curve suggests that bonding starts with the onset of a continuous gradual transfer of electrons in the a system from Ti+ to N with a concomitant continuous small transfer (back bonding) of electrons in the u system from N to Ti'. At the equilibrium Ti-N separation of 3.253 au this results in 0.26 electron transferred in each a bond and a delocalization of 0.12 electron from the N 2pu into the Ti+ 3du, 4s, and 4p in the u system. The N 2s remains doubly occupied throughout. The total
-
+
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 2989
Metal Nitride Cations 41! MCSCF 411 MCSC,\::
V+ 5F asymptote V+ 5D o s y m p t o t 4
0-
2040
-
h
I
E
v
0
0
60-
80 MCSCF 100-
2A MCSCF+ 1 +2 120-
VNf
R(V-N)
(OU)
Figure 12. Electron population of selected atomic orbitals of u and a symmetry from the valence natural orbitals of the 76 CSF MCSCF wave function and gross Vt to N charge transfer of triply bonded *A VN' as a function of internuclearseparation. Rq designates the calculated bond length. Note: only one of the two identical a's is pictured.
(411) and *,a-doubly bonded (42-)species have dissociation energies less than half of that for the 2A VN', being 39.1 and 34.5 kcal/mol, respectively. C. Analysis of the MCSCF Wave Function along the Potential Curve. a,?r,a-Triply Bonded VIP. For triply bonded VN+ the atomic occupations of the a and one set of the identical a atomic orbitals and the gross V+ to N charge transfer obtained from the MCSCF NOS for the "5D character conserving" MCSCF calculation, which dissociates to the S C F SFlimit for V+, as a function of internuclear separation are shown in Figure 12. This plot is similar to that for CrN+ above 5.00 au, as would be expected, because the S C F state is 3dN for both bare metal ions. For the ?r orbitals the plot is like that for CrN+ with gradual Vt to N charge transfer in the ?r system beginning when the separation is less than 3.75 au. For the a orbitals, although it is like that for CrN+ below 3.50 au, the plot shows an obvious blip in the V+ 4s and 3da and N 2pa populations at a separation of 4.25 au, the same distance at which the M+ 3da and N 2pa bonding interaction becomes important for CrN+ and TiN+. However, the adiabatic and "5D character conserving" potential curves do not show any kinks near this V-N distance. Although with this MCSCF ansatz the pure "SF4s bonding" and pure "5D 3da bonding" MCSCF potential curves would be shown to cross at this point, this is an
artifact of the MCSCF ansatz and not a real crossing and is not physical. Between 5.00 and 3.50 au the character of the a populations and the V+ to N charge transfer in the a system is obviously intermediate between that of CrN+ and TiN+. We should keep in mind that this MCSCF scenario is predicated on the accurate S C F representation of the V+ states, both in the isolated ion and in situ. Of course, for VN+ the most usefully accurate description of the electronic structure by Mulliken population analysis at all V-N distances could have been obtained only if the differential correlation between the 5D and SFstates of V+ had been included at the MCSCF level so that it would dissociate to the 5D limit for V+. We expect a population plot based on such a MCSCF calculation to be similar to that for CrN+ and believe that the 4s-3da mixing around 4.25 au is solely due to the differential correlation energy error in the 3d4 configurations causing perturbation by 4s. However, the original population analysis is useful since there remains no doubt that the N 2s does not participate at any V-N distance nor any doubt that at the equilibrium bond length of 2.954 au the a bond is totally between the V+ 3da and the N 2pa, with no contribution from the V+ 4s. Again, at equilibrium the V+ to N charge transfer in the a system (0.1 1 electron) is the same as that in each of the a systems (0.12 electron). The gross V+ to N charge transfer is a maximum (0.38 electron) at equilibrium. a,?r-Doubly Bonded VN* and a,a-Doubly Bonded V W . The V-N internuclear distance dependencies of the atomic populations of the a and a NOS and gross V+ to N charge transfer (not shown) obtained from the MCSCF NO'S in a,a-doubly bonded VN+ (411) and in a,r-doubly bonded VN+ (42-)are similar to the equilibrium portions of the plots for the corresponding TiN+ species. The features in the population plots for the a,a-doubly bonded VN+ are remarkably similar to that for the triply bonded VN', the differences being greater V+ to N charge transfer in the a and A systems and the ?rx back donation. At the equilibrium bond length of 3.369 au the V+ to N charge transfer in the a system (0.21 electron) is the same as that in the aysystem (0.20 electron), but this condition is reached at longer bond lengths than for the triply bonded species. For the a,?r doubly bonded VN+ at the equilibrium bond length of 3.260 au the Vt to N charge transfer is 0.26 electron in each a system. For both doubly bonded VN+ species the total V+ to N charge transfer (including the back bonding) is a maximum (0.37 and 0.35 electron, respectively) at equilibrium and is the same as for the triply bonded VN+ at equilibrium. ScN+. A . MCSCF Wave Function Construction. a,*-Doubly Bonded S c I P . The wave functions for a,?r-doubly bonded ScN+ (211, 2B2)is analogously constructed from that for TiN+ 3BI by removing the spectator 3dii electron. The spectator electron in the N 2p, orbital gives a doublet. The MCSCF function separates + N(4S)] where a(M) is 4s. to the S C F products [SC+(~D) a,a-Doubly Bonded S c W . The wave function for a,*-doubly bonded ScN+ 22+(2A1)is constructed in a similar manner from that of TiN+ 3A by removal of the spectator 3d6 electron. The spectator electron in the N 2pa gives a doublet. Since the ground-state Sc+ 3D does not contribute, the 3d2 3Fexcited state is the Sc+ asymptote. The MCSCF function separates to the SCF + N(4S)J. products [SC+(~F) a-Singly Bonded S c W . The high-spin ScN+ molecule formed by spin pairing an electron on Sc+ in the (3d14s1)3D or (3d2) 3F state with an electron on 4S N to allow one a bond is a 4A state (4A, in C2J. The asymptotic form of the GVB representation of the wave function is as follows: ScN+: 4A (4A1)
-
[a(M)a(N)
+ a(N)a(M)] 3d6- 2p,
2p,
where M = Sc and the electrons in the N 2py and 2p, and Sc+ 3d6- orbitals are coupled into a quartet. The MCSCF function of this form permits all spin couplings among the five valence + N(4S)] electrons and separates to the S C F products [SC+(~D) where a(M) is the 4s. a-Singly Bonded ScN+. The high-spin ScN+ molecule with one A spin pair between Sc+ in the (3d2) SF excited state and N in the 4S state is calculated with 4B1 symmetry in C2". The
2990
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
Kunze and Harrison
19
.-c
"I
om
211 MCSCF M,C:;l+Z
'frl
1
BO J
I
lool
00
I
,
20
,
I
,
40
R(SC-N)
,
I
60
,
80
,
9
loo
02
(OJ)
Figure 13. Potential energy curves for the u,a-doubly bonded 211 state and the a-singly bonded 4A state of ScN' as a function of internuclear separation for the MCSCF and MCSCF+ 1+2 calculations.
1
$+ ', ;Y',
Req
00
00
1
2;
Sc+ , d
20
40
R(Sc-N)
EO
60
100
(au)
Figure 15. Electron population of selected atomic orbitals of u and a symmetry from the valence natural orbitals of the 17 CSF MCSCF wave
-404
function and gross Sc' to N charge transfer of a,a-doubly bonded zII ScN' as a function of internuclear separation. R, designates the calculated bond length.
-2oj
O i
,Oi
"1
i ScN+
100
'
o - oo2
0
R(Sc-N)
Figure 14. Potential energy curves for the a,r-doubly bonded z2' state and the a-singly bonded 4B,state of ScN' as a function of internuclear
0.0 0.0
ScN':
-
(4B1) [ay(M)ay(N) + ny(N)ay(M)] 3d6- 2pa 2p,
where M = Sc and the extra electrons in the N 2pa and 2p, and Sc' 3d6- orbitals are coupled into a quartet. The MCSCF function of this form permits all spin couplings among the five valence + N(4S)]. electrons and separates to the S C F products [SC+(~F) B. Potential Curves. Shown in Figure 13 and 14 are the MCSCF and the MCSCF+1+2 potential curves for the four ScN' species relative to the sum of the ground-state fragments. Of the ScN+ species studied, the one of lowest energy is predicted which has a dissociation to be the a,a-doubly bonded species (22+), energy of 63.1 kcal/mol with respect to the ground-state fragments and 81.5 kcal/mol with respect to the excited-state asymptotic fragments at the MCSCF+1+2 level. The u,a-doubly bonded (2n)species has a very similar dissociation energy of 57.3 kcal/mol. The u-singly bonded (4A) species and the a-singly bonded (2BI)species have dissociation energies of 29.2 and 22.2 kcal/mol, respectively. C. Analysis of the MCSCF Wave Functions along the Potential Curves. a,a- Doubly Bonded ScW. The M-N internuclear distance dependencies of the electron populations in the ci and a atomic orbitals and the gross charge transfer obtained from the MCSCF NOS in u,a-doubly bonded ScN' are shown in Figure 15 and are similar to that for u,n-doubly bonded TiN+. The main differences are that for ScN' the features occur at a slightly longer bond length and the Sc' to N charge transfers are slightly larger in the ci and R, systems and the av back donation. The distance dependence of the MCSCF N O occupation numbers for the u orbital5 and the R orbitals for ScN+ are shown in Figure 8 and
,
,
,
1
Re 2.0
4.0
R(Sc-N)
separation for the MCSCF and MCSCF+1+2 calculations. asymptotic form of the GVB representation of the wave function is as follows:
,
0 . 2 1 sc+,,,+,
(au)
6.0
8.0
10.0
(au)
Figure 16. Electron population of selected atomic orbitals of u and a symmetry from the valence natural orbitals of the 17 CSF MCSCF wave function and gross Sc' to N charge transfer of n,s-doubly bonded z2' ScN' as a function of internuclear separation. R, designates the cal-
culated bond length. are similar to that for the a,a-doubly bonded TiN'. Again, at the equilibrium bond distance of 3.402 au the Sc' to N charge transfer in the CT system (0.28 electron) is the same as that in the R, system (0.29 electron). The small ryback donation (0.09 electron) reduces the total charge transfer in the A system, making the a system the more polar. The total Sc+to N charge transfer is a maximum (0.49 electron) at equilibrium. a,n-Doubly Bonded S c w . The M-N internuclear distance dependencies of the electron populations in the a and a atomic orbitals and the gross charge transfer obtained from the MCSCF NOS in a,n-doubly bonded ScNS are shown in Figure 16 and are similar to that for the a,a-doubly bonded TiN'. The main differences are that for ScN+ the features occur at a slightly longer bond length and the charge transfers are slightly larger. The distance dependencies of the MCSCF N O occupation numbers for the a orbitals for ScN' are shown in Figure 10 compared to TiN' and are similar except that the divergence is greater. At the equilibrium Sc-N separation of 3.31 1 au, this results in 0.30 electron transferred in each a bond and a back bonding (0.13 electron) from N 2pa to, the Sc+ do, 4s and 4po. The total Sc' to N charge transfer reaches a maximum (0.49 electron) at the equilibrium bond distance. a-Singly Bonded Sew. The Sc-N internuclear distance dependencies of the electron populations in the a atomic orbitals and the gross Sc' to N charge transfer obtained from the MCSCF NOS in u-singly bonded ScN' are shown in Figure 17. The distance dependence of the MCSCF N O occupation numbers for
The Journal of Physical Chemistry, Vol. 93, No. 8. 1989 2991
Metal Nitride Cations
1.8
0.61
'A
ScN'
00 00
20
40
80
60
R(Sc-N)
'B 1
\
100
00
20
4.0
R(Sc-N)
(OU)
Figure 17. Electron population of selected atomic orbitals of u and A symmetry from the valence natural orbitals of the 6 CSF MCSCF wave function and gross Sc' to N charge transfer of u-singly bonded 4A ScN' as a function of internuclear separation. R, designates the calculated bond length.
60
.o
(0.)
Figure 19. Electron population of selected atomic orbitals of u and n symmetry from the valence natural orbitals of the 6 CSF MCSCF wave function and gross Sc' to N charge transfer of *-singly bonded 'B, ScN' as a function of internuclear separation. R, designates the calculated
bond length.
20
C
.-0
1.5-
a 3 u u
0
-
c 0 ._
e 0
1.0-
-
-
0
L
2
0.5-
Z
0.0, 00
x-Singly Bonded ScN+. The Sc-N internuclear distance dependencies of the electron populations in the u and x atomic orbitals and the gross charge transfer obtained from the MCSCF NOS in x-singly bonded ScN' are shown in Figure 19. The distance dependence of the MCSCF N O occupation numbers for the a orbitals for ScN+ are shown in Figure 18, where they can be compared to that of the u orbitals for u-singly bonded ScN'. They are similar to that for the x orbitals in the doubly bonded ScN+ species except that divergence is greater at longer Sc-N separation, reaching the same numbers at longer bond lengths. When the Sc-N separation is less than 5.00 au and the bonding N O occupation numbers are 1.4, the potential curve suggests that bonding starts with the onset of a continuous gradual transfer of electrons in the a, system from Sc+ to N with a concomitant continuous small transfer (back bonding) of electrons in the u and r ysystems from N to Sc'. At the equilibrium Sc-N separation of 3.729 au this results in 0.62 electron transferred in the axbond and a delocalization of 0.09 electron from the N 2pu and 2s into the Sc' 3du, 4s, and 4p in the u system and 0.03 electron in the r ysystem. The total Sc' to N charge transfer reaches its maximum (0.53 electron) at the equilibrium bond distance. Again, this is the same magnitude as for the other ScN+ species.
I
I
2.0
- I
I
4.0
R(SC-N)
,
I
6.0
I
8.0
10.0
(au)
Figure 18. Occupation numbers for the natural orbital pair corresponding to a formal bond of u or A symmetry for the 6 CSF MCSCF wave functions of a-singly bonded 4A ScN' and n-singly bonded 4B1 ScN', respectively, as a function of internuclear separation. R, desig-
nates the calculated bond length. the u orbitals ScN+ are shown in Figure 18. This species differs from the others in this study in that the formal bond has 4s character as well as 3da character. The N O occupation numbers have significantly diverged when the Sc-N separation is decreased to 6.00 au, but the potential curve has only a slight slope and the singly occupied orbitals remain singly occupied. When the occupation number of the u bonding N O is 1.4 at 5.00 au, Sc' 4s to N 2pu charge transfer begins, and the potential curve indicates a very shallow well. It is not until 4.50 au that the potential energy begins to drop as the Sc' 3du orbital starts to become populated with further depletion of the 4s. The atomic populations suggest that the u bond of the Sc-Nf u-single bond is the most polar of those studied. The N O occupation numbers diverge most rapidly for the u system between 5.00 and 4.50 au. At the equilibrium bond length of 3.946 au, the Sc' contribution to the u bond has 4s as well as 3du while the N bond orbitals and the lone pair show 2s-2pu hybridization. The total Scf to N charge transfer has a maximum (0.54 electron) at the equilibrium bond distance. This is the same magnitude as for the doubly bonded ScN' species.
-
Results and Discussion Summary of Results. Various early-transition-metal nitride formal u,a,a-triple, u,a-double, n,x-double, mingle, and a-single two-electron bonds have been studied by ab initio MCSCF and multireference CI techniques. The potential energy curves, the plots of the distance dependencies of the atomic populations of the NOS and of the N O occupation numbers, and Tables I-IV present the main energetic and spectroscopic results and characterization of the equilibrium MCSCF wave functions for the variously bonded CrN+, VN+, TiN+, and ScN+ molecules studied, including the equilibrium bond lengths and bond energies, the intrinsic bond energies, and other spectroscopic information. The important features are the following: within each class of multiple-bonded species the bond lengths are remarkably constant; the formal u bonds in the multibonded species have exclusively M' 3du and N 2pu character; bonding in all states has significant ionic M2'N- character due to charge transfer from Mf to N ; the gross M+ to N charge transfer is constant for all the nitrides studied of a given metal +1 cation, despite different numbers of formal bonds; the equilibrium bond lengths all occur at the in-
+ 1 cations with
2992
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
Equilibrium Electronic Structural Predictions. Table 111 summarizes information characterizing the equilibrium MCSCF wave functions for all species, including Mulliken populations, gross M+ to N charge transfer, and natural orbital (NO) occupation numbers. While the limitations of the Mulliken population analysis6 preclude detailed arguments based on them, we are confident of the following: any 2p electron on N that is not spin-paired with a M+ electron is delocalized slightly into the empty M' orbital (back bonding); at equilibrium, the a populations of the formal u bonds in the multibonded species have all evolved into 3da with no 4s or 4pa, despite the dN-'4s1 ground-state configuration of some of the metal +1 cations; only the a-singly bonded ScN+ 4A has 4s character in a formal bond. This means that the Ti-N and Sc-N multiple bonds are solely between unhybridized N and the 3dN (4F and 'F) excited states of Ti+ and SC'. The half-covalent and half-ionic nature of the bonding is evident. For each species the M+,N spin pairs correspond to polar covalent bonds with significant M+ to N charge transfer. Only at equilibrium is the M' to N charge transfer equally distributed over all formal bonds. The equilibrium bond lengths all occur at the internuclear distance of maximum M+ to N electron charge transfer. Remarkably, regardless of the number of bonds in the various MN+ species, the gross M+ to N charge transfer is constant for the nitrides of a given metal +1 cation, as if it depended only on the properties of the metal such as the second ionization potential (IP) and the nitrogen electron affinity. Further, as one would expect from the second IPSfor Sc, Ti, V, and Cr (297.3, 31 3.4, 327.2, and 380.2 kcal/mol, re~pectively),~ the average metal-to-nitrogen charge transfer at equilibrium increases somewhat in the order CrN+ (0.33) < VN+ (0.37) < TiN+ (0.44) < ScN+ (0.50). Ionic Character Considerations. Consider the separation, R, where the Coulombic attraction interaction energy (potential energy) of a q+ = +2 point charge for a q- = -1 point charge, q+q-/R,equals the second IP. This distance is R = 2.232, 2.1 17, 2.027, and 1.746 A for Sc, Ti, V, and Cr, respectively, and clearly differs dramatically with the different metals. However, for each metal, all MN+ bond lengths, even the single bonds, of the species studied are shorter than these critical lengths. The charge transfer starts at distances longer than these critical points but in the order consistent with these critical points. Thus, even simple ionic point charge interaction can be an important factor in determining the bond length and of course the bond strength. The strength of the bonds is partly due to ionic contributions from M+ to N charge transfer which, in a gross manner, precludes the participation of the 4s orbital except for Sc+ because the spatial extent of the 4s orbital results in it bonding at longer bond distances than the critical lengths.
TABLE 11: Metal Nitride +1 Cation First Anharmonicity Constants ( w s cm-I), Rotational Constants ( f i n em-'), Vibration-Rotation Constants (uc, c d ) , and Centrifugal Distortion Constants ( d e , lod cm-') from the MCSCF+l+Z Level Calculation' WeXe
MN' 'A2 2A2 'A,
CrN+ VN' TiN'
'2zA '2'
VNt TiN' ScN+
4rI
VN' TiN' ScN'
42'A 2Z'
4Al
ScN'
'A
4A,
2rI
Triple 8.2 6.6 5.4
4
Pe
Bonds 0.600 0.620 0.619
0.0063 0.0046 0.0038
1.161 0.944 0.867
MN' a,r-Double Bonds 4B2 8.4 0.508 0.0049 3Bl 6.1 0.502 0.0041 zBz 5.2 0.486 0.0035
0.837 0.748 0.694
MN' n,s-Double Bonds 7.8 0.540 0.0044 'A2 5.8 0.536 0.0039 2Al 4.8 0.522 0.0032
0.960 0.826 0.750
MN+ &ingle Bonds 4.9
0.362
MN' &ingle Bonds 4Bl 3.9 0.406
ScN'
0.0034
0.602
0.0027
0.574
MCSCF+ 1 +2* level for CrN'.
ternuclear distance of maximum M+ to N electron charge transfer; only at the equilibrium bond lengths is the M+ to N electron charge transfer equally distributed over all formal bonds; the dissociation energies when referenced to the ground-state fragments vary widely for a given bond type, but when augmented by an estimate of the exchange energy that was lost upon bond formation and referenced to the 3dN M+ state are remarkably constant. Equilibrium Nuclear Structural Predictions. We will focus on MCSCF+1+2 level for our discussion. That the metal-nitrogen triple-bond length in CrN', VN+, and TiN+ is nearly constant (- 1.585 A) with a spread of only 0.022 A suggests that there is a characteristic early-transition-metal-nitrogen triple-bond length. Although the metal-nitrogen a,*-double-bond and a,*double-bond lengths increase slightly in the order VN+ < TiN+ < ScN+, the are also remarkably constant (averages 1.767 and 1.709 , respectively), with spread$ of only 0.064 and 0.054 A. The bond lengths of excited-state C H (42-)triply bonded to Cr+, V+, and Ti+ were also predictedId to be constant with an averge of 1.758 8, and a spread of 0.014 A, 0.173 A longer than the triply bonded MN+. The ScN' a-single bond at 2.088 A is longer than the *-single bond at 1.973 A, but overall, the bond lengths become shorter with increasing bond order. The vibrational frequencies, however, vary for a given bond type.
-
- x
Kunze and Harrison
TABLE 111: Characterization of the Metal Nitride +1 Cation MCSCF Wave Functions at Equilibrium
-
pa
px
py
gross Mt charge transfer (electron)
1.11 1.11 1.13
1.10 1.12 1.14
1.10 1.12 1.14
0.76 0.06 0.07
MN' a,n-Double Bonds 1 1 2.00 1.21 1 0 2.00 1.26 0 0 2.00 1.28
0.92 1.25 1.29
0.09
0.76 0.70 0.65
0.76 0.70 0.65
MN' n,n-Double Bonds 1 1 2.00 0.90 1 0 2.00 0.88 0 0 2.00 0.87
0.25
0.04
0.04
valence confign M'
IRR
N
MN'
state
C2,
s
do
dn,
dry
CrN' VN' TiN+
'22A
3Az 2A2
0.00 0.00
1 1
0
'A,
0.00
0.87 0.84 0.80
0.87 0.84
'Z+
0.86 0.86 0.86
0.80
0
0
VN' TiN+ ScN'
4rI
0.00
2rI
4Bz .'B, 2B,
0.76 0.71 0.68
0.05 0.70 0.64
VN' TiN' ScN'
423A 22+
4A, ]A2 2Al
0.00 0.00
0.07
0.00
ScN+
4P
4A,
0.15
d6-
d6+
s
bonding N O occupation no. U
iT
0.33 0.38 0.43
1.84 1.89 1.91
1.80 1.87 1.90
1.20 0.92 0.91
0.37 0.44 0.49
1.84 1.88 1.91
1.80
1.21 1.26 1.30
1.21 1.26 1.30
0.35 0.44 0.49
1.58
0.95
0.95
0.54
MN' r-Single Bonds 1 0 1.93 0.95
1.62
0.96
0.53
MN+ Triple Bonds
0.00 0.00
0.08
1
2.00 2.00 2.00
1.86 1.90 1.79 1.83
1.89
MN' a-Single Bonds
ScN+
4Bl
0.01
0.08
0.29
0.03
1
0
1.97
1.96 1.95
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 2993
Metal Nitride Cations
TABLE IV: Metal Nitride +1 Cation Bond Energies (kcal/mol) Relative to the Ground-State Fragments (and Relative to Excited State AsvmDtotic Fragments) and Intrinsic Bond Enerdes (De*,kcal/mol)" Relative to the M+ High-Spin dN States ~
MCSCF CrNt VN+
'z-
VN+
4n
TiNt
2A
'A2 2A2
'2+
TiN+
Kdd"
AKdda
De*
99.OC
17 15 15
77 44 22
127 131 123
56.OC 55.3
15 15 14
37 22 7
76 81 83
15
37
72
15
22
82
14
I
89
0.6 19.7 (32.6) 33.9 (60.7)
M N + *,*-Double Bonds 33.3 34.5 50.1 54.3 (59.8) (60.0) 63.1 (81.5)
54.6 (61.3) 63.0 (81.5)
4AI
7.1
M N + &ingle Bonds 27.0 29.2
26.8
14
7
55
4BI
-0.3 (26.5)
M N + *-Single Bonds 19.7 22.2 (46.5) (42.7)
21.1 (39.6)
14
7
50
2n
2B2
VN+ TiNt
42-
'A
4A, 'A2
ScN'
2z+
2Al
ScNt
MCSCF (N2S)+1+2
M N + a,*-Double Bonds 38.1 39.1 60.2c 55.V 57.3
'B1
4A
+
5.5 27.3d 29.1
4B2
ScN+
ScN+
34.3 66.8 80.0b
MCSCF MCSCF 1+2* +1+2 M N + Triple Bonds 49.5 86.0 87.1 97%
"De* = De + AKdd + E,, where D, = dissociation energy at MCSCF+l+2 (MCSCF+1+2* for CrN') level, E, = promotion energy to the d"' state, Kdd = average exchange integral for dNstate of M+ (kcal/mol), AKdd = exchange energy loss of M+ from formation of bonds (kcal/mol). bRelative to (4/5)1/2(4F)- (1/5)1/214P)Ti+ and 4S N fragments. cRelative to the molecular asymptote at R = 20.0 au. dRelative to (3/5)1/214F) + (2/5)'/214P) Tit and 4S N fragments.
Energetics. Calculated Bond Energies. In Table IV we report the dissociation energy, De, of the variously bonded CrN', VN', TiN+, and ScN+ at the various theoretical levels referenced to ground-state M+,N as described for the potential curves. Since the 2Z+ and 4B, states of ScN+ and the 3D state of TiN+ separate to M+ in the dN excited state and N in the ground state, we also report the calculated De referenced to the asymptotic fragments. Note that all states studied are bound with respect to the ground-state fragments. For the species within a given bond type, the calculated De vary widely, decreasing drastically in the order from ScN' to CrN+. For instance, the triply bonded TiN+ ( I Z ' ) has a De twice that of the triply bonded CrN+ (3Z-). Also the *,*-doubly bonded species for ScN+ (*E+), which dissociates to the excited-state Sc+, has a larger bond energy (relative to the ground-state Sc+) than the triply bonded CrN+ (3E-). The triply bonded TiN+ is the most strongly bound of the molecules in this study. Interestingly, the constancy of the bond lengths in each of the types of multibonded metal nitride +1 cations is not borne out by a similar constancy in the calculated bond energies. Intrinsic Bond Energies. It has been suggested by us and othersl-2dv*that when a high-spin metal forms a bond, the calculated bond energy is the stabilization energy that remains after payment of the differential intraatomic exchange energy associated with uncoupling pairs of like-spin electrons on the metal. Ab initio SCF calculations of the exchange integrals for the lowest 3dNstate of M+, the state in situ (3d5 6SCr+, 3d4 5DV+, 3d3 4 F Ti', and 3d2 3F Sc'), lead to an estimate of the exchange-energy loss upon bond formation for each metal for the various bond orders, reported in Table IV. We estimate the intrinsic metal-nitrogen bond energies, De*, as the sum of the calculated De and the calculated exchange-energy loss, referenced to the asymptote with the 3dN configuration of the transition-metal ions. These De* are reported in Table IV for the MCSCF+l+Z level (MCSCF+1+2* level for CrN'). Thus we find that the similarity of the bonding is also reflected in the approximate equality of the metal-nitrogen intrinsic bond strengths for the triply bonded M N + species (- 127 kcal/mol), the u,x-doubly bonded species (-80 kcal/mol), and the n,n-doubly bonded species (-81 kcal/mol). That the "intrinsic" bond strengths are in the intuitively anticipated order is shown in Figure 20 of the intrinsic bond strength vs formal bond order. The nonlinearity of this plot is similar to that for N-N bonds, as can be seen from the dashed line giving the ratio of the N-N bond energy9 to the M-N+ De* for each bond order. Re-
2.0
t
1404
*.
120-
.-C
a 5
01
?
al
n
.
1.5+
z I
100-
I 0
a
80-
D
.
2
604
._ V
.
-
40-
1 . O d
z I
z c
VI
.E -c
0
0.5.9 c 0
1
E
0.0
Bond Order
Figure 20. Comparison of the intrinsic bond energies and the bond order for the variously bonded CrN', VNt, TiN+, and ScN' species. Ratio (dashed line) of the N-N bond energies and average M-N intrinsic bond energies as a function of the bond order.
markably, the contribution is similar for s and u bonds. Unlike bonding in main-group elements, the u and T bonds appear to have equal strength, Le., the u bonds are weak. While this simple model brings some order to the bond strengths, it must be emphasized that the calculated and not the augmented bond energies should be compared with experiment after correction for zero-point energy. Predictions of Chemistry. To predict chemistry, one starts by using a simple Hess's law addition using the data from this work, the available MR+ data in the and average maingroup bond energiesg However, unlike main-group chemistry, adjustments are necessary to take into consideration that the Sc+ and Ti+ 3dN-I4s' to 3dN promotion energy has already been overcome in M N + and that the differential exchange-energy debt has already been paid since the electrons have already been encumbered in bonds in MN+. Recall that the M-N * bonds are (8) Carter, E. A.; Goddard, W. A., 111 J . Phys. Chem. 1984, 88, 1485. (9) Cotton, F. A.; Wilkinson, G. Basic Inorganic Chemisfry;John Wiley & Sons: New York, 1976; pp 8-11.
2994
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
TABLE V: Number of Configuration State Functions (CSFs)O
CrN+ ' 2 VN+ 2A TiN+ I.2'
M C S C F MCSCF MCSCF MCSCF +1+2* +1+2 (N2S)+1+2 M N + Triple Bonds 'A2 126 134432 (234952) (600338) 2A2 76 76303 119783 (332376) IA, 37 31423 100015 4B2 'BI 'B2
MN' a,n-Double Bonds 34 33606 83330 39176 25 23888 17 14926
(235166) 121377 54156
VN' 42- 4AI TiN' 'A 'A2 ScN+ 2Z+ 2A,
M N + n,r-Double Bonds 34 33174 82612 25 23663 38915 17 14899
(235146) 121 102 54444
VN+ 411 TiN+ SCN+ 2rI
ScN+ 4A
4A,
M N + a-Single Bonds 6 5105 8752
32862
ScN'
4B,
M N + &ingle Bonds 6 4970 8680
32684
"Numbers in parentheses indicate that the calculation was too large for our facilities.
as favorable as the M-N a bonds. Both are weak relative to N-H bonds and comparable to MH+ bonds. When these are considered and we then note that there is M+ to N charge transfer, we find that the following predictions are suggested: While only Sc+ of the early-transition-metal + I cations will exothermically break the H-H bond in H 2 or a C-H bond in hydrocarbons, such bonds will be broken by exothermic addition across a M-N bond of the ground-state, multiply bonded MN+, where M = Sc, Ti, V, or Cr, to form R'MNR', where the fragments R' and R are either hydrogens or alkyl groups. The more electrophilic fragment should preferentially attack at the N. Furthermore, the linear form of the R'MNR' species will be the more stable product since the M-R' and N-R bonds prefer to be mingle bonds, while the M-N bonds have no preference
Kunze and Harrison and can be a,a-double bonds for M = Ti, V, and Cr or a a-single bond for M = Sc. This would not be expected in main-group chemistry where H N N H is bent in a zigzag manner. We also note that the ground-state a,*-doubly bonded ScN+ *Pwith an unpaired electron in the N 2pu should readily scavenge a hydrogen atom or alkyl radical to form linear ScNR'.
Conclusion These calculations on the early-transition-metal nitride + 1 cations, CrN+, VN', TiN+, and ScN', with formal triple, g,adouble, a,a-double, mingle, and a-single bonds permit the following conclusions: 1. The determining factors for the bond length are (1) the bond order, (2) the requirements for the charge transfer, and (3) the spatial extent of the M+ du and d n orbitals (rather than 4s) along the bond axis. The observations that lead one to such a suggestion include the following: Bond Order. (a) Within each class of bond type for the multibonded M N + molecules studied the bond lengths are remarkably constant. (b) Bond length decreases with increasing bond order, regardless of which orbitals are used for the formal bonds. Requirementsfor the Charge Transfer. (a) Bonding in all states has significant ionic M2+N-character due to charge transfer from M+ to N. (b) Most importantly, the gross M+ to N charge transfer as a function of internuclear distance corresponds to a maximum at the equilibrium bond distance. (c) For all bond classes only at the equilibrium bond length is the charge transfer equally distributed over all formal bonds. (d) Charge transfer is associated with a sharp drop in the potential energy curves for those species that dissociate to the dN state of the metal ion. For those species that dissociate to the 4s'3dN-' state of M+, the sharp drop in the potential energy curves is not associated with charge transfer from the M+ 4s but occurs only after the 4s charge is sufficiently depleted and the 3da has begun to be occupied, with a net charge transfer from the M+ 3du to the N 2pg.
TABLE VI: Total Energies (hartrees) of the Metal Nitride +1 Cations at the Equilibrium Bond Length, R,, and at the Molecular Asymptote, R = 20.0 au
CrN'
3 2-
'A2
VN'
2A
2A2
(d4) (sd') TiN+
'2'
VN'
4rI
TiN'
4B2 'Bl
MCSCF MCSCF +1+2* M N + Triple Bonds -1097.568 89 -1097.636 26 -1097.51430 -1097.557 10 -997.166 17 -997.232 74 -997.051 I O -997.095 25 -997.059 75 -997.091 45 -902.706 25 -902.578 86 MN' a,n-Double -997.068 54 not done -902.622 36 -902.578 86 -8 13.975 6 1 -813.92925
Bonds -997.14426 not done -902.696 84 -902.600 93
MN' *,*-Double -997.058 80 not done -902.621 56 -902.569 64 -8 13.983 28 -8 13.886 52
Bonds
MCSCF +1+2
MCSCF (N2S)+ 1+2
-997.245 50 -997.106 11 -997.097 93 -902.773 61 -902.61781
-902.852 95 -902.695 23
-997.168 72 a -902.706 67 -902.6 17 8 1 -8 14.048 69 -8 13.957 42
-902.78446 -902.695 23 -8 14.126 91 -814.038 77
-997.16 1 48 a -902.708 3 1 -902.61222 -814.057 93 -8 13.924 57
-902.790 21 -902.692 41 -8 14.139 26 -814.00288
ScN'
2rI
2B2
VN'
42-
4Al
TiN+
'A
'A2
ScN+
22+
2Al
ScN+
4A
4A,
M N + &ingle -8 13.940 59 -813.92925
Bonds -813.99988 -813.95688
-8 14.003 96 -8 13.957 42
-814.081 5 5 -8 14.038 77
4Bl
MN' *-Single Bonds -8 13.988 23 -8 13.928 69 -813.88652 -813.914 15
-813.99281 -813.92457
-814.07252 -814.00288
ScN+
-997.13655 not done -902.698 14 -902.602 74
"The elimination in Ch symmetry becomes contaminated at long bond distances
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 2995
Metal Nitride Cations TABLE VII: Fragment Total Energies (hartrees)
N state 4s
IRR C2,
valence confign
4A~
S2P0'P,'Py'
IRR M+ Cr+ V+
c2c
state 6S
6A1C
6D
6A2C
Sct
SCF+1+2
SCF(N2S)+l+2
-54.40074
-54.428 38
-54.509 80
v aIen ce
confign dddr,'dny'd8-d6+' s'da'd?rx'dry'db+'
SCF
SCF+l+2*
-1043.1 13 54 -1043.076 29
-1043.12905" -1043.09634"
-942.650 09 -942.650 34
5D
same
5BI 5AIC 5BI 4AlC 4A2 4B2 4AIC 3AIC 'BI 'A2 3AlC 'A2 'B2
5F
Tit
SCF+I +2*
SCF
4F (4/5)1/214F)- (1/5)1/214P) (3/5)1/214F)+ (2/5)1/214P) 4F
3D 3F
-942.658 99
same
-942.655 -942.667 -942.655 -942.663 -942.659
09' 30' 46" 25b 37"
SCF+ 1+2 SCF(N2S)+ 1+2 -1043.154 58 -1043.096 34 -942.678 15 -942.678 40
same -942.669 79
same
-848.18947 -848.178 10
-848.18989'
-848.193 41 -848.19309
same
-848.178 59' -848.174 52b
same
-848.16888 -759.528 -759.528 same -759.494 -759.485 same
75 49 24 76
-759.52849' -759.494 24' -759.485 76'
-848.18428 -759.529 30 -759.52906 same -159.499 65 -159.499 60
same
8 orbital kept singly occupied. 'The real orbital occupancy yields the correct spatial symmetry as "Two 6 orbitals kept singly occupied. determined in Hay, P. J. J . Chem. Phys. 1977, 66, 4377.
TABLE VIII: Metal +1 Cation Promotion Energies (kcal/mol) ground-state excited-state M+ confign state confign state 6S d4s' 6D Cr+ d5 5D d's' 5F V+ d4 4F Ti* d2s' 4F d3 3D d2 'F sc+ d's' ~~
~
exptl" 35.1 7.8 2.5 13.7
SCF' 23.4 -5.6 12.9 21.7
SCF+1+2' 36.5 5.2 5.7 18.6
" Moore, C . E. Aromic Energy Levels; Nat. Stand. Ref. Data Ser., Nat Bur. Stand., Cir. 35; Washington D. C., 1971; Vol. I and I1 (averaged over values for J states for each term). 'Calculated under C , symmetry by using energies from Table VI1 for the calculation using the real orbital occupancies that yield the correct spatial symmetry as determined in Hay, P. J. J . Chem. Phys. 1977, 66, 4377. (e) The gross M+ to N charge transfer is constant for all the nitrides studied of a given metal +1 cation, despite different numbers of formal bonds. This appears to depend only on the properties of the metal such as the second ionization potential and the nitrogen electron affinity, increasing somewhat in the order expected from the second ionization potentials for Sc, Ti, V, and Cr. (f) The ScN+ mingle bond shows 4s-3da mixing, but the M+ component of the formal u bonds of the other species is exclusively 3do. Stabilization by ionic contributions due to M+ to N charge transfer would preclude the significant participation in the bonding of the 4s orbital because the spatial extent of the 4s orbital places most of the 4s density at longer bond distances than the separation where the simple ionic MZ+,N-point charge Coulombic attraction energy equals the second ionization potential for each M. For each metal all MN+ bond lengths of the species studied, even the single bonds, are shorter than these critical lengths. Spatial Extent of the Occupied Orbitals along the Bond Axis. (a) The bond lengths of the u,a-doubly bonded species are longer than that of the a,n-doubly bonded species. Within each type of double bond there is a correlation between the bond lengths and the N 2pa population whether in the formal bond or in N to M+ back bonding. (b) The ScN+ a-single bond is longer than the n-single bond presumably because the ScN+ mingle bond has Sc+ 4s character. 2. All states studied are bound with respect to the ground-state fragments. The ground states are the triply bonded species for CrN+ (32-),VN' (*A), and TiN+ ( I Z ' ) and the n,a-doubly bonded species for ScN+ (*E+),although the a,n-doubly bonded species for ScN+ (*n)is only 6 kcal/mol higher. The triply bonded TiN+ is the most strongly bound of the molecules in this study. 3. The dissociation energies referenced to the ground-state fragments vary widely for a given bond type due to different
exchange-energy losses upon bond formation for the M+, but when augmented by an estimate of the exchange energy loss and referenced to the 3dN M+ state, the resulting intrinsic bond dissociation energies are remarkably constant. Thus, unlike bonding with main-group elements, the u and a contributions to bond strength appear to be equal, Le., the u bonds are weak. 4. Within the context of these calculations, we expect that the n,a-double bonds will persist in MNR+ species, where R is a u-singly bonded species. Acknowledgment. We are indebted to the Argonne National Laboratory Theoretical Chemistry Group for providing the QUEST-164 electronic structure codes used in this study. This work was supported in part by the National Science Foundation (Grant No. CHE85 19752). Appendix: General Computational Details Basis Sets. The basis sets for the M+ have been described previously.' They consist of 14s,l lp,6d primitive Gaussian functions constructed by augmenting Wachter'sIo 14s,9p,5d basis with two additional diffuse p functions" and an extra d as recommended by Hay.12 The M+ functions were contracted to 5s,4p,3d by using the method for generalized contraction due to Raffenetti.I3 The basis set for the nitrogen was devised by Almlof and Tay10r.I~ They augmented the (1 3s8p) primitive set of van Duijneveldt15 to produce a (1 3s8p6d) primitive Gaussian basis (10) Wachters, A. J. H. J . Chem. Phys. 1970, 52, 1033. (1 1) Dunning, Jr., T. H., private communication. (12) Hay, P. J. J . Chem. Phys. 1977,66, 4377. (13) Raffenetti, R. C. J . Chem. Phys. 1973, 58, 4452. (14) AlmlBf, J.; Taylor, P. R. J . Phys. Chem. 1987, 86, 4070. (1 5 ) Duijneveldt, F. B. IBM Technical Research Report No. RJ-945; IBM Research Laboratory: San Jose, CA, 1971.
2996 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 set and contracted it to [5s4p2d] using their atomic natural orbital (ANO) general contraction scheme. The contraction consists of the coefficients of the ANOs with the largest occupation number obtained in an atomic single S C F reference configuration singles and doubles CI calculation with these uncontracted primitive functions. Molecular Codes. All molecular calculations were done on a FPS- 164 jointly supported by the Michigan State University Chemistry Department and the Office of the Provost, using the Argonne National Laboratory collection of QUEST-1 64 codes. In particular, the integrals were calculated with the program ARGOS written by Pitzer;I6 the S C F and GVB initial estimate calculations used the GVB164 program by Bair;I7 the S C F and MCSCF final calculations used the UEXP program and related utility codes written by Shepard.18 The configuration interaction calculations were done with the program UCI (and its related utility codes) written by Lischka et aLI9 MCSCF Wave Function. Structural Correlation and SpinCoupling. The bond dissociation energy calculations involve all-electron ab initio wave functions for the molecular cations and the corresponding fragments. All MN' calculations involved starting with a multiconfiguration self-consistent field (MCSCF) wave function that provides structural correlation of the bonds in a generalized valence bond (GVB) fashion' and includes all possible spin couplings among the valence electrons consistent with the desired spin state, including that of the spectator electrons in the singly occupied orbitals with the bonding electrons. The construction of the wave function for each state studied is described earlier. The MCSCF function is constructed and solved variationally as a function of the M-N separation and dissociates properly to the sum of the single self-consistent field (SCF) configuration functions for the fragments in their correct spin state (the molecular asymptote). Dynamical Correlation. Dynamical correlation for the eight (CrN+), seven (VN+), six (TiN'), or five (ScN') electrons in the MCSCF active space was included by using multireference (MR) all singles and doubles (SD) configuration interaction (CI) (MRSDCI). Several MRSDCI calculations were carried out, each using the entire MCSCF wave function as reference and each designed to dissociate properly to the corresponding level single S C F reference configuration (SR) all singles and doubles C I (SRSDCI) calculation on the atoms. Our experience] is that these calculations are size-consistent,20 Le., at each theoretical level, the molecular asymptotic energies are equal to the sum of the energies of the atoms. For all calculations, no excitations were permitted from the Ar core of M+ or the 1s core of N. First, we denote as MCSCF+1+2 the MRSDCI calculation with S D excitations from all valence orbitals (no excitations were permitted from the N 2s orbital) in all MCSCF configurations. This calculation correlates the electrons involved in the bonds as well as the spectator electrons in the M+ 3d and N 2p singly occupied orbitals. It dissociates to the SRSDCI calculation, denoted SCF+1+2,for each fragment with the singly occupied 4s and 3d orbitals on M+ correlated and the 2p orbitals on N correlated. Second, we denote as MCSCF(N2S)+l+2 a MRSDCI calculation where the doubly occupied N 2s core orbital is also included in the active space for each configuration in the MCSCF. This dissociates to the SCF+1+2 calculation for M+ and, for N, the SRSDCI calculation in which the N 2s is also active, denoted as SCF(NZS)+1+2. Since for CrN+ the MCSCF+ 1+2 calculation from the 126 configuration MCSCF reference space was too large for our computational facilities, we also used a third, more limited, MRSDCI calculation, denoted as MCSCF+I +2*, that correlates (16) The ARGOS integral program was developed by R. M. Pitzer (Ohio State University). (17) The GVB164 program was written by R. Bair (Argonne National Laboratory). (1 8) A dewription of the UEXP program is given in: Shepard, R. J . Chem.
Kunze and Harrison the cr and ?r electrons but keeps the d6 orbitals at the MCSCF level. It contains the configurations present in the MCSCF+1+2 calculation except that the singly occupied Mf d6 orbitals were kept singly occupied. This is reasonable, since the d6 orbitals are spatially out of the way of the bonding region and we have used it before with success.1c The asymptote is the N atom at the SCF+ 1+2 level and the M+ atom calculated by using a SRSDCI calculation with the'singly occupied d6 orbitals not correlated, denoted as SCF+1+2*. Spectroscopic Constants. Spectroscopic constants are obtained from the potential curves by means of a Dunham analysis.21 A fourth-order polynomial, fitted to five points around the minimum, was used to determine equilibrium bond length and vibrational frequency (Table I) at the various theoretical levels, and other spectroscopic constants at the MCSCF+1+2 level (Table 11; rotational constant, vibrational frequencies, centrifugal distortion constant, vibration-rotation interaction constant, and first anharmonicity constants). Computational Results. For each molecule discussed the number of spin eigenfunctions or configuration state functions (CSF)19 in C2, symmetry for the various theoretical levels is tabulated in Table V. For each theoretical level used the molecular absolute energies at the equilibrium internuclear distances and at an internuclear distance of 20.0 au are collected in Table VI. The corresponding atomic absolute energies are collected in Table VII. These data show that all procedures dissociate properly in a size-consistentmanner to within a millihartree, except for the CI calculations for the 2A and 3Bl states of TiN', where there is a 4 mhartree difference due to a symmetry mixing under C2, symmetry. For each state, at equilibrium the dominant configuration in the MCSCF calculation would be the S C F result and has substantial weight, since there are no near-degeneracy effects.8 This is also reflected in the N O occupation numbers, which are significant only for the bonding partner of each pair. The MCSCF+ 1+2 potential curves start to drop in energy at larger distances than do the MCSCF curves but have the minimum at nearly the same distance. This suggests that the relative contributions of the M+, M2+, N, N-, and other states to the M-N bonds are well described by the MCSCF wave functions at equilibrium but not as well at longer bond distances. Generally, however, except for the V+ S C F asymptote being 5D, instead of SF,the relative MCSCF energies track the MCSCF+ 1+2 energies over the whole curve reasonably well, and we are confident that the essential physics is in the MCSCF formulation. For a given species the inclusion of correlation results in a substantial increase in the calculated De, but the D,'s are essentially constant for the different levels of correlation treatment performed here. Thus, the MCSCF+1+2* and MCSCF(N2S)+1+2 potential curves are too close to that for MCSCF+1+2 to be seen and are not displayed. The calculated bond lengths are not very sensitive to the level of calculation, but with the addition of CI the triple bond lengths increase and that for the other bonds decrease. The data suggest that not correlating the triplet-coupled electrons in CrN+ results in the Cr-N bond being slightly longer (-0.003 A) than the "fully" correlated calculation would predictz2 The calculated vibrational frequencies are very sensitive to the level of calculation. With the addition of CI the triple-bond frequencies decrease and that for the other bonds increase. The MCSCF+1+2* and MCSCF(N2s)+ 1+2 calculations give nearly the same frequency results as the MCSCF+ 1+2 calculation. Reliability of the Computations. Since the molecular wave functions can involve a significant mixing of the low-lying Mf asymptotes, one measure of the quality of the calculation is how well the atomic splittings are reproduced. Table VI11 collects the experimental (averaged over values for J states for each term)5 and S C F and SCF+l+2 level calculated energy separations for
Phys. 1982, 76, 543.
(19) Lischka, H.;Shepard, R.; Brown, F. B.; Shavitt, I. Int. J . Quantum Chem. Symp. 1981, 15, 91.
(20) Bartlett, R.J. Anuu. Reu. Phys. Chem. 1981,32, 359, and references therein.
(21) Dunham, J. L. Phys. Reu. 1932, 41, 721. (22) Also the CrN+ 0,and bond length results from the present calculation are in reasonable agreement with our previouslE MCSCF+ 1 +2 calculation with a slightly smaller basis set, which consisted of 249 208 CSFs.
J . Phys. Chem. 1989, 93, 2997-2999 the high-spin states of (Ar core)3dN-'4s' and (Ar core)3dNconfigurations for Sc+, Ti+, V+, and Cr+. Note that at the SCF level the energies of the two states are in the experimental order for Cr', Ti+, and Sc+ but not for V+, where the SF(3d34s1)state is more stable. Therefore, we include the SFV+ energies in Table VII. Work2d*23*24 suggests that a reasonable calculation of the relative energy of a first-row transition element or small molecule containing such an element is possible if one at least includes the differential electron correlation between the low-lying states of different configurations of both the transition-metal and ligand (23) Botch, B. H.; Dunning, Jr., T. H.; Harrison, J. F. J. Chem. Phys. 1981, 75, 3466.
(24) Walsh, S . P.; Bauschlicher,Jr., C. W. J. Chem. Phys. 1983, 78,4597.
2997
fragments. Because d-d radial correlation up to the singles and doubles level is included in the MCSCF+ 1+2 calculations (all occupied d orbitals are in the active space), the SCF+ 1+2 energies are in the correct order for all the M+ states. Comparison of our previous work with experiment has allowed us to assess the absolute accuracy of our calculations.lc*d When calculated from the MCSCF+1+2 wave functions, our De's tend to be 25% low. Thus, dissociation energies calculated from the MCSCF+1+2 wave functions should be lower bounds and will likely increase with calculations that include a more extensive level of correlation. Bond lengths are expected to be more reliable than our energetic predictions. Registry No. ScN+, 119145-01-6; TiN+, 83018-06-8; VN', 11020720-0; O N + , 83017-97-4.
Ground-State Geometry of the (+C,H,)V and (T~-c,H,)v+Half-Sandwich Complexes by Local-Spin-Density Linear Combination of Atomic Orbitals Techniques Saba M. Mattar* and William Hamilton Department of Chemistry, University of New Brunswick, Bag Service No. 45222, Fredericton, New Brunswick, Canada E3B 6E2 (Received: July 14, 1988; In Final Form: August 23, 1988)
The electronic structures of (v6-C6H6)v and (&C6H6)V+ have been computed by an LCAO method using two different local density functionals. The neutral molecule and the cation are predicted to have stable bound ground states with an optimal benzene-vanadium distance of 3.06 and 3.08 au, respectively. The dissociation energies required to separate the V and C6H6 are found to be in the range of 5.25-3.05 eV. They are of the same order of magnitude as those predicted from thermodynamic data.
In the past few years, molecular beam experiments have been reported where various metal atoms or metal dimers and organic ligands have resulted in benzene-metal products.'S2 These experiments provide us with dissociation energies and heats of formation of the fundamental interactions between the metal atoms and the C6H6 moiety. The beam experiments produce complexes that are short-lived due to their high reactivity. It is desirable to have complementary theoretical computations that confirm/ predict the equilibrium geometries and binding energies of these products. These computations would also illustrate the structure-bonding relationships of these complexes. Ab initio electronic structure computations of such molecules, using large basis sets and including electron-electron correlation effects, are very scarce because of the large expense in computer time and costs. Recently, the study of the 550-nm absorption band formed by the co-condensation of v atoms with C6H6/Ar mixtures, at 12 K, led to the discovery of the matrix-isolated (v6-c6H6)v halfsandwich c o m p l e ~ . ~Its , ~structure was confirmed by electron paramagnetic resonance (EPR), infrared, and UV-visible spectroscopies. A self-consistent-field scattered wave (SCF-SW) computation of its electronic structure, using the X a approximation, was also p e r f ~ r m e d .The ~ geometry was assumed to be that of bis(benzene)vanadiums with one C6H6 ring r e m ~ v e d . ~ There are no reports to date of the existence of (v6-C6H6)V in the gas phase. It is important to understand the electronic structure, bonding, and properties of (?f-C&,)V because the v atom has a naked
hemisphere to which the addition of other metal atoms or stabilizing ligands (eo,c s , NO, C6H6, etc.) is possible. It is thus an important reactive intermediate to be used in the synthesis of a variety of organometallic complexes and clusters. A fundamental question regarding (v6-c6H6)v comes to mind: is this molecule stable in the gas phase, or does it exist because the C6H6 and v fragments are simply held together and isolated by the matrix at 12 K? The answer may be obtained by computing the total energy of the complex as a function of the benzenevanadium distance and searching for a geometry that displays a bound ground state. In such a case the SCF-SW method is inappropriate for geometry optimization because the change of sphere radii with different geometries renders the comparison of the total energies meaningless. Here we report the geometry optimization and the electronic structure of (v6-C6H6)v using a local-spin-density linear combination of atomic orbitals (LSD-LCAO) method that gives accurate total energies.6 The electronic structure was computed by using the Xa approximation (a= 0.7). In a separate treatment the effects of electron exchange and correlation were introduced using the interpolation formulas of Perdew and Zunger' (PZ approximation) that parametrize the exchange-correlation potentials and energy densities of an electron gas as computed by Ceperly and Aldere8 The 13~/7~/5d[4333/43/41+]basis set of Andzelm et al. for V was used.g This basis set is specifically designed for LSDLCAO computations. A 9s/3p[3s/2p] for canbon and a 3s[2s] for hydrogen were employed for the C6H6 ring." The auxiliary
( I ) Hettich, R. L.; Freiser, B. S.J. Am. Chem. SOC.1985, 107,622245226. (2) Whetten, R. L.; Cox, D. M.; Trevor, D. J.; Kaldor, A. Surf. Sci. 1985, 156, 8. (3) Andrews, M. P.; Huber, H. X.;Mattar, S. M.; McIntosh, D. F.; Ozin, G. A. J. Am. Chem. SOC.1983, 105, 6170-6172. (4) Andrews, M. P.; Mattar, S.M.; Ozin, G. A. J. Phys. Chem. 1986, 90,
744-753. (5) The C6H6-V distance was that of (C6H6)>Vtaken from: Meutterties, E. L.; Bleek, J. R.; Wucherer, E. J.; Albright, T. A. Chem. Reu. 1982,82,499.
0022-3654/89/2093-2997$0 1.50/0
(6) (a) Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. J. Chem. Phys. 1979, 71, 4993. (b) Lamson, S.H.; Messmer, R. P. Chem. Phys. Letf. 1983, 98, 72. (7) Perdew, J. P.; Zunger, A. Phys. Rev.B 1981, 23, 5048. (8) Ceperly, D. M.; Alder, B. J. Phys. Reu. Lett. 1980, 45, 566. (9) Andzelm, J.; Radzio, E.; Salahub, D. R. J. Compur. Chem. 1985, 6, 5 20-5 3 2.
0 1989 American Chemical Society
