Heisenberg model for radical reactions. Part 3. Direct exchange coupling between transition metal ions and triplet methylene

THEO CHEM ELSEVIER

Journal of Molecular Structure (Theochem) 310(1994) 185-196

Heisenberg model for radical reactions. Part 3. Direct exchange coupling between transition metal ions and triplet methylene S. Yamanaka'':", T. Kawamura", T. Noro", K. Yamaguchiv'' a Department of Chemistry. Faculty of Science, Hokkaido University, Sapporo 060. Japan bDepartment of Chemistry. Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan

(Received 19July 1993; accepted 12 November 1993)

Abstract Ab initio UHF calculations have been carried out for naked transition metal-methylene ions. The low-spin open-shell states are found to be the ground states of these ions, in accord with the GVB-CI results. The low-spin transition metalmethylene ions are regarded as direct exchange-coupled systems formed from high-spin transition metal ions and the triplet monocentric diradical. The direct exchange couplings between these species can be described on the basis of the Heisenberg model. The CASSCF calculations using UHF natural orbitals (UN Os) have been carried out for low-spin states, showing that UNOs and their occupation numbers are good trials. However, they provide poor binding energies for the transition metal-carbon double bonds and indicate the necessity for dynamical correlation corrections. The approximate spin-projected UHF Moller-Plesset and UHF coupled-cluster methods provide reasonable binding energies for the transition metal-methylene ions, which are close to those of modified coupled-pair functional (CPF) and average CPF methods.

1. Introduction Armentrout and co-workers [1-3] have shown that naked transition metal-methylene molecular ions are produced by the following gas-phase reactions: M+

+ CH 4 ---+ M=CHi + H 2

+ c-C3H6 ---+ M=CHi + C2H4 M+ + c-C 2H 40 ---+ M=CHi + CH 20 M+

(Ia) (Ib) (Ic)

They have determined experimentally the metalmethylene bond dissociation energies, D(M+-CH 2 ) . • Corresponding author.

Judging from the large dissociation energies, the metal-methylene bond could be regarded as a double bond; both the a and 7r bonds are singletpaired, in conformity with the low-spin ground state. However, Carter and Goddard [4] have shown that the high-spin (6B1) state of the chromium-methylene ion is calculated to be more stable than the low-spin (4Bd state when the ab initio open-shell restricted Hartree-Fock (RHF) method is adopted. In other words, the RHF SCF theory is inadequate for describing the weak -r-bonding between Cr+ (S = 5/2) and CH 2 (S = I). The GVB perfect-pairing (PP) calculation is in qualitatively good agreement with experiment. However, it failed to reproduce the greater stability

0166-1280/94/S07.00 © 1994 Elsevier Science B.V. All rights reserved SSDJ 0166-1280(93)03615-E

186

S. Yamanaka et alp. Mol. Struct, (Theochem) 3/0 (1994) 185-196

of the low-spin state (4B1) . Schaefer and co-workers [5] have also shown that the 7B1 state is calculated to be the ground state of Mn=CH! by RHF and RHF SD CI calculations, although the experimental results indicate the ground sB 1 state. Similarly, the high-spin ground state was predicted for the naked nickel-methylene complex by RHF calculations [6]. Thus, the nature of the 7r bond for naked metal-carbene systems appears to be very different from that of such carbene complexes, satisfying the 18-electron rule as (CO)sCr=CXX', for which the closed-shell RHF descriptions seem to be reasonable [7-10]. In a previous paper [11], we have shown that the above failure of the SCF calculations can be removed when the double occupancy constraint for the singlet pairs is relaxed; thus, the unrestricted Hartree-Fock (UHF) MO method correctly reproduces the greater stability of lower spin states for M=CHi+ compared with corresponding higher spin states. Ab initio MO calculations indicate an important role of exchange (spin) polarizations for 7r bonds in stabilizing the low-spin states of these naked metal systems. The binding energies (BEs) between transition metal ions (M = Cr+ and Mn+) and triplet methylene have been calculated by the approximately projected UMP method using the TZ plus polarization (TZP) quality. The values calculated by APUMP4 are found to be close to those of MR CIjjCASSCF reported by Harrison and coworkers [12-14], PMP4 by Mckee [15] and GVB CI by Carter and Goddard [4]. However, they are only one-half of the experimental BEs first presented by Armentrout and co-workers [1-3]. However, Armentrout et al. [2] have reported recently new experimental values for M=CH! which are considerably smaller than the previous ones. Our previous values [II] for Cr=CHl (44.3kcalmol- l ) and Mn=CH! (60.0kcalmol- ) correspond to 82 and 85% of the new experimental values (53.8 and 70.6kcalmol- I), respectively. During the progress of the present work, Bauschlicher et al. [16] have also published the CASSCF and extensive MR CIjjCASSCF calculations and elucidated the theoretical BEs between M+ and CH z on the basis of the highest level of the spin-restricted approach. According to their

internally contracted averaged (lCA) coupled-pair functional (CPF) results, the BEs are 48 and 57 kcal mol-I for the Cr=CH! and Mn=CH! ions, respectively. These values are comparable with those of APUMP4 [11]. In this paper, we first examine direct exchange couplings between transition metal ions and triplet methylene on the basis of spin-UHF, approximate projected UHF (APUHF) and UNO CASSCF methods. Relative stabilities among the low-, intermediate- and high-spin states are calculated at the UHF and APUHF levels. The BEs are calculated on the basis of the post-UHF methods: APUMP4 and UHF CC SD(T) in particular. The simple Heisenberg model is also presented in order to reproduce direct exchange coupling energies for the M=CH! ions.

2. MO-theoretical description of direct exchangecoupled systems

2.1. Different orbital for different spins (DaDS) The ab initio UHF calculations of the transition metal-methylene ions were carried out as described previously [11]. The basis sets used for the transition metals are the Tatewaki-Huzinaga MINI-I and MIDI-I [17] supplemented by the 4p AO with the same exponent as that for the 4s AO, and by Hay's diffuse orbital [18]. The MINI-I and MIDI-I plus diffuse orbital sets were employed also for the C, 0 and H atoms. The former (MINI plus diffuse) and latter (MIDI plus diffuse) basis sets will here be referred to as BSI and BSII, respectively. TheJ-polarization function was also added to the preceding TZ basis set (BSII), giving the TZP basis set (BSIII). The M -C bond distances for the transition metal-methylene ions were determined by energy optimization at the UMP4 (BSIII) level for semiquantitative discussions of the BEs. Figure I illustrates the coordinate axes for the M=X n+ systems. The bonding, non-bonding and antibonding a orbitals are constructed from the 2p: AO of X, and the nd:2, (n + I)s and (n + l)p: AOs of the ansition metal M, as illustrated in (A) of Fig. 2. These are denoted as ¢(a), ¢(an) and ¢(a*),

S. Yamanaka el

sur. iIIol. Struct.

x Y

(Theochem} 310 (1994) 185-196

187

(X = CH z) were first calculated to elucidate the orbital energy gap (~Eorb) between the bonding and antibonding a or 71 orbitals since ~Eorb is responsible for each bond strength. The HS UHF solutions for such states are generally written by these bonding and antibonding MOs:

i M=C-"'-"---c~ z

"H

Fig. I. Coordinate axes for the transition metal-methylene systems.

p+4 WUHF(UU*, 7171*)

= 1
(2) respectively. The bonding and antibonding n-M'Os,
where e, denotes the remaining open-shell orbitals and K signifies inner-shell orbitals which are less spin-polarized. The energy gaps for the a and 71 MOs of BCrCH!, for example, are 6.2 and 3.5 eV by UHF (BSI), respectively. Because of these small energy gaps, both the a- and rr-bcnding MOs in the LS states for M=X"+ are spin-polarized as expressed approximately with the orbitals of the HS states:

'l/J+(u) = cos(w/2)¢(u) ± sin(w/2)¢(u*)

(3a)

l/J+(n) = cos(w/2)
(3b)

where the magnitudes of orbital mixing parameters (w) are dependent on the corresponding orbital energy gaps (~forb)' The orbital overlap between l/J+(q) and l/J-(q) is given by Tq = (l/J+(q)Il/J-(q)} = cos w(q) (q = o or n)

(4)

The Tq-value for an unstable bond q should be far smaller than 1.0 as shown below. The UHF noMOs with such small T, are more or less localized on either of the metal ion and carbon sites as illustrated in Fig. 3. Therefore, the UHF MOs are similar to the GVB orbitals in this case, although ,,'-(n*)

9(0*)

\V+(1t*),,/

Ijf(n)

~

(A)

~

(8)

~

(e)

Fig.2. Orbital correlation diagrams for the (7. and lI"·bond formations from the transition metal ions and triplet methylene. (A) and (B) denote, respectively, the stable a- and ,,·orbital formations, while (C) denotes the unstable ,,·orbital formation.

S. Yamanaka et al.lJ, Mol. Struct, (Theochem} 3/0 (1994) 185-196

188

Table I States and orbital configurations for the ground states ofisoelectronic transition metal-methylene ions and configurations of transition metal ions States

lA, 2A I 2A I 2A, 3n; 3

AI 3A2

d/s m a

d2 d 2s1 d 2s1 d3 d 5s 1 d 3s1 d~

~A,

d~

4n; 5n l 4n,

d5 d 5s 1 d 6s d 1s d 8s d 8s d lo

4B~

2 A1 2AI IA I

a

b

Orbital configurations"

,

Systems

a

11"

11"

a

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

, 1I".L

61

0 0 0 0 I 0 0 0 I I I 2 2 2 2

0 0 I 0 I 0 I I I I 2 2 I 2 2

o o o I

o I I I I I I I 2 2 2

o I

o o

o I

o I

o I

o o 2

I 2

SccH1+. TiCH~+ ScCH 2 TiCH1+ ZrCH1+, VCH~+ VCH1+ TiCH 2, ZrCH 2 NbCH1+, CrOW VCH2, NbCH 2 CrCH1+' MocHl+, MnCHi+ CrCH 2, MoCH 2, MnCH1+, FeOW FeCH1+ CocHl+ NiCH1+ (case I) NiCH1+ (case 2) CucH1+

Configurations of transition metal ions. a: a .. 11"6: b .. a': a .. To': bj , 1I".L: b 2• 61: a .. 62: a2. an: a,.

the spin-coupling is not restricted to the perfectpairing (PP) type. Then the orbital correlation diagram for the 1I"-bond formation is given by (C) of Fig. 2. The LS UHF solutions are given by these leftright split orbitals: P'lJ UHF( G)

= 17/1+ (a)tp- (a) 7/1+ (11")1,&- (11" )7/lp (K ) I (5)

~

~ I/(lt)

~

~

Fig. 3. Left-right split methylene ions.

1ji(n')

11"

P'lJRHF(ZW)

= 17/1- (a)7/I-(a) 7/1- (11")7/1- (1I")7/l p (K ) I (6)

This latter expression accounts for the reason why the RHF solution tends to predict the LS state as being the less stable for naked transition metal-X (X = CH 2 , NH, 0) systems. The situation should entail RHF CI calculations involving a large number of multiple excitations for reasonable descriptions of the LS state, as has been discussed by Vincent et al. [5]. The intermediate spin (IS) states are similarly given by

1jI-(lt)

~~ Ij/(n')

The corresponding RHF solution should converge to the ionic high-energy state M+ 2X-2 (X = CH 2) , which is expressed approximately as

orbitals for transition metal-

P + 2'lJ UHF(1m') =

17/1+ (a) 7/1- (a)4>( 11")4>(1I"')7/lp (k) I

(7a) p+ 2'lJ UHd aa') = 14>(a) 4> (a' )7/1+ (11")7/1- (1I")7/lp (k)I

(7b) where 11"7i' and co" denote the 11"11"' and ao" excited states, respectively. Thus the HS, IS and LS radical states of M=CH! can be described within the different orbital for different spins (DODS) approxi-

S. Yamanaka et 0/./1. Mo/ . Struct , (Thcochem} 3/0 (/994) /85-/96 Table 2 Calculated energy difference between the highest spin (H5) and lowest spin (l5) states of the chromium-methylene monocation CrCHi Method

~EIlS-LS

Ga

(kcalrnol") RHF b

GVB pp b UHF APUHF GVB CI

-53.7 -11.3 15.1 19.3 19.0

6B, 6B\ "B\ "B 1 "B,

Predicted ground state. Ref. 4. C This work (B511). a

b

mation, whereas the zwitterionic (ZW) state is described by the RHF solution. 2.2. Antiferromagnetic exchange-coupling between transition metal ions and triplet methylene 2.2.1. Chromium and manganese methylene ions The r.-7r' excitation energy values of the lowspin (4BI) state for Cr=CH! calculated by several methods at a fixed interatomic distance R(Cr-C) = 1.9 A are compared in Table 2. The excitation energy defined as the energy difference between the HS (6BI) and LS (4BI) state of Cr=CH! ~EHS-LS =

E(HS) - E(LS)

(8)

is negative in sign when it is calculated by the RHF SCF method [4], as has already been mentioned. The situation remains the same at the GVB PP method [4], which does involve pair correlation, but certainly does not involve exchange correlations arising from the spin-flip excitations of radical electrons. The difficulty in the GVB approach lies in the fact that the PP approximation is too restrictive to describe exchange couplings between open-shell transition metals and high-spin organic molecules such as CH 2 , NH and O. The UHF (BSII) solution, however, which involves both correlations approximately, correctly predicts the greater stability of the LS state, as does the GVB RCI method [4]. The energy gap by APUHF is close to that of GVB RCI [4]. The results indicate that both pair and spin correlations play an important role in the stabilization of the 4B) state of Cr=CH! .

189

Table 3 summarizes the total energies of various states, excitation energies (in parentheses), and net charges on Cr and C calculated for Cr=CH 2+ (n = 0-2) by the UHF SCF procedure. As can be seen in Table 2, the (aa'), (r.,,') and (",,', aa') excitation energies for CrCH 2+ (n = 0-2) are all positive in sign. The trend is independent of the basis sets (I and II) employed. However, the magnitudes of the excitation energies are sensitive to the basis set used as well as the formal charge. The net charges on the Cr and C atoms are positive and negative in sign, respectively, in all the states examined here. From the net charge populations and the -r-bond splitting (Fig. 3), the Cr-C tt bond could be regarded as a resonance hybrid of the covalent (DR) and zwitterionic I (ZW-I) structures. The polarization of the Cr-C bond is similar to that in Schrock-type alkyl carbene complexes R 3Ta=CRR' [19], in which Ta has 10d electrons. The contribution of the other (ZW-II) structure is small in this case . . + + Cr-C(DR) Cr- C (ZW-I) Cr-C (ZW-II)

in contrast to Fischer-type carbene complexes; (CO)sM=CORR' (M = Cr, Mo and W) [20-22]. Mn=CH 2+ (II = 1 and 2) has also been treated by the UHF (BSI) method. The results are shown in Table 3, where it can be seen that the low-spin 6- n B. state is the ground state, in contrast to the open-shell RHF and RHF SO CI methods [5]. The net charges on the Mn and C atoms are positive and negative in sign, respectively, in both the ground and tm" excited states. Polarization of the Mn-C bond is in line with that of the Schrock-type manganese carbene complex Cp(COhMn=C(CH 3h [20-22]. The closed-shell RHF solution 1AI responsible for ZW-I is far more unstable compared with the LS open-shell state in the case of M=CH 2+ (M = Cr, Mn, II = 0-2) as shown in Table 3. The additions ofligands to the naked cores are essential for stabilization of the closed-shell structures in Schrock-type carbene complexes [19]. 2.2.2. First and second transition metal-methylene ions Some of the first and second transition metal

S. Yamallaka et al.l J. Mol. Struct, [Theochem} 310 (1994) 185-196

190

Table 3 Total excitation energies (in parentheses] and net charges for chromium-methylene systems CrCH~+ System

CrCHr

State

Orbital configurations

,

a

1r

1r

a'

SBI(G) 7BI(u') 7BI (all") 9BI (1r1r' ,aa' ) IAI(G)

2

2

o

2

I

I

1 1

2 I

o

o o

2

2

o

4B I (G)

2

2

o

I

I I

o o

Net charges

I I I I 2

I I

I I 2

I I I I

I I I I

0

0

o

o

2

2

o

o

o

o lBl(G)

2

2

o

sB1(r.,,'}

2

1

I

o

2 I

o

L I

2 2 I 2

2

2 2

2

° °o °o oo

I I

I I

SRI(ca'} 7BI (r.;;-', uu') MnCHr

SRI(G)d ')

1 B I (u

9 B I (m!",

aa')

IAI(G) 4B I (G )

6BI (u') 8BI (r.lr', all")

I

I I

2

I

I I

o

I

I I I 2

o I

I I I 2

-108 I. 74123 (0.0) -1081.7214 (12.5) -1081.6075 (83.9) -1081.6225 (74.5) -1081.5168 (140.7) -1081.4985 -1081.5293 -1081.4571 -1081.5052 -1081.4341 -1081.4595 -1081.4559 -1081.4760

(0.0) (O.O)e (26.0) (l5.I)e (40.4) (43.8)< (29.5) (33.5)<

Cr

C

0.292 0.400 -0.191 -0.284 0.647

-0.594 -0.704 -0.230 -0.121 -0.912

0.390 I.I31 0.996 1.242 0.75& 0.796 0.776 0.9~0

-0.302 -0.416 -0.407 -0.531 -0.245 -0.169 -0.250 -0.231

o o o o

0 0 Q 0

-1081.8783 (0.0) -1080.8298 (30.5) -1080.8190 (37.3} -1080.8292 (30.9)

1.530 1.571 1.593 1.695

-0.113 -0.168 -0.207 -0293

I I I 0

I I I 0

-1187.9077 -1187.8932 -1187.7836 -1187.7552

(O.O} (8.7) (77.5) (95.3)

1.009 1.108 0.540 1.011

-0.429 -0.511 -0.050 -0.433

o o o

-1187.3531 (0.0) -1187.3100 (27.0) -1187.3145 (24.3)

1.505 1.523 1.645

-0.099 -0.131 -0.247

Values in parentheses in kcal mol-I. [5333/5353/5] for Cr, [31/31] for C and [3] for H (BSI). e [53321/5321/411] for Cr, [211/211] for C and [21] for H (BSII). d R(Mn-C) = 1.95 A. a

b

ions in Table 1 were examined in order to confirm the results for the M=CH;+ ions (M = Cr, Mn) in the preceding section. Table 4 summarizes the total energies of various states, excitation energies (in parentheses), and net charges on M and C calculated for first and second transition metalmethylene ions by the UHF (BSI) method. As can be seen in Table 4, the (7l"iT"'), «(T(T*) and (1riT"*, all") excitation energies for these ions are all positive in sign. The magnitudes of the excitation energies arc dependent on the metal ions and their formal charges. Thus antiferrornagnetic direct exchange couplings can be qualitatively described even at the UHF BSI level, in contrast to RHF and GVB PP methods.

3. UNO CASSCF calculations of the ground state In order to elucidate the nature of the double bonds between the transition metal ions and triplet methylene, the UHF natural orbitals (UNO) analysis has been carried out for the ground states. Table 5 summarizes the occupation numbers (lI q (p» of the bonding and antibonding a and 1r orbitals for transition metal-methylene ions obtained by the APUHF (BSIII) method. Since the Tq values for dz--pa bonds are in the range 0.2-0.3, the magnitude of the radical character y is intermediate (40-60%) as in the case of the rr bond of 1,3-dipoles such as ozone. The y value is

S. Yamanaka et 01./1. Mol. Struct, (Theochem) 310 (1994) 185-196

191

Table 4 Total energies, excitation energies (in parentheses) and net charges for first transition metal-methylene ions' Systems

Net charges

States

2A1{G) 4A1(r.r.') 4A1(aa') 6A t{r.r.·aa·) IAI{G) SccHi 3AI (r.r.') 3A I{aa') 5A I{r.r.'aa') 3AI{G) TiCH 2 5A. (n') 5A.{aa·) 7A.{r.r.'aa') 2A.{G) TiCHi 4AI(r.r.') 4At (aa') 4A t{r.r.'aa') 4A2{G) VCH 2 6A2{n ' ) 6A 2{aa') 8 A 2 (1iiT· aa·) 3A2{G) VCHi 5A 2{r.r.') 7A 2{r.r.'aa') FeCH~+ 5B.{G) 7B.(r.r.') 7B. (aa') 9B1(r.r.'aa') ScCH 2

-798.2052 (0.0) -798.1732 (20.1) -798.0847 (75.6) -798.0360 (106.2) - 798.0029 (0.0) -797.9639 (24.5) -797.9114 (57.4) -797.8771 (78.9) -836.3530 (0) -886.8283 (15.5) -886.7174 (85.1) -886.6821 (107.2) -886.6360 (0.0) -386.6046 (19.7) -886.5305 (66.2) -886.5293 (67.0) -981.3037 (0.0) -981.2816 (13.9) -981.1644 (87.4) -981.1618 (89.0) -981.0709 (0.0) -981.0384 (20.4) -980.9982 (45.6) -1299.9310 (0.0) -1299.8447 (54.2) -1299.6327 (249.9) -1299.5151 (261.0)

Systems

M

C

0.328 0.318 0.501 0.407 1.247 1.337 1.060 0.883 0.314 0.345 0.395 -0.205 1.226 1.305 0.954 0.864 0.303 0.376 0.017 -0.274 1.023 1.247 0.803 1.397 1.602 0.478 0.518

-0.643 -0.646 -0.938 -0.040 -0.618 -0.712 -0.531 -0.381 -0.634 -0.674 -0.834 -0.234 -0.613 -0.695 -0.460 -0.363 -0.622 -0.700 -0.455 -0.156 -0.431 -0.639 -0.294 -0.055 -0.194 0.619 0.521

States

Net charges M

3A.{G) 5A. (n') 5A.{aa·) 7A.{r.r.'aa') 2A1{G) ZrCHi 4A1(r.r.') 4A t{aa') 6A t{r.r.·aa·) NbCH 2 4A2{G) 6A 2{r.r.·) 6A 2{aa') 8 A 2{r.r.'aa') NbCHi 3A2{G) 5A2{n ' ) 5A 2{aa') 7A 2{r.r.· aa') MoCH 2 5B,(G) 7Bt (r.r.') 7BI(aa') 9B I{r.r.'aa') MaCHi 4B1{G) 6B. (r.r.') 6B, (aa') 8Bt{r.r.'aa') ZrCH 2

-3574.7072 (0.0) 0.319 -3574.6406 (41.8) 0.365 -3574.5914 (72.7) 0.240 -3574.4923 (134.8) -0.182 -3574.4983 (0.0) 1.258 -3574.4312 (42.1) 1.342 -3574.3862 (70.3) 1.143 -3574.3301 (105.5) 1.002 -3789.1249 (0.0) 1.672 -3789.0729 (32.6) 1.720 -3788.8885 (148.3) 1.423 -3788.8315 (184.1) 1.131 -3788.9222 (0.0) 2.164 -3788.8650 (35.9) 2.253 -3788.7799 (89.3) 2.157 -3788.7680 (976.8) 2.005 -4010.9256 (0.0) 0.313 -4010.8808 (28.1) 0.440 -4010.7476 (111.7) -0.015 -4010.7290 (123.4) -0.146 -4010.6927 (0.0) 1.100 -4010.6377 (34.5) 1.215 -4010.5816 (69.7) 1.031 -4010.5698 (77.1) 1.028

C -0.679 -0.727 -0.704 -0.284 -0.678 -0.760 -0.687 -0.553 -1.018 -1.073 -0.877 -0.585 -0.603 -0.690 -0.716 -0.558 -0.645 -0.776 -0.441 -0.299 -0.524 -0.632 -0.564 -0.546

= 1.91 A. Values in parentheses in kilocalories per mole.

a R{M-C) b

defined by y = 1 - 2Tq / (l

+ T~) =

Il q

(antibonding)

(9)

In contrast, the Tq-valuesfor the da-pa bonds are almost 1.0, showing the weak radical character. Judging from the occupation numbers, the remaining bond orbitals are the closed-shell type. Thus the four-electron four-orbital subsystem [4,4] can be extracted for the UNO CASSCF calculations. Since the M=CHI systems have singly occupied molecular orbitals (SOMa), these are also included, leading at least to the [4 + 111,4 + Ill] CAS space, where III denotes the number of SOMas. The UNO CASSCF calculations have been carried out in order to clarify the bonding natures

in the M=CHI ions. The occupation numbers of natural orbitals by the UNO CASSCF calculations are shown in Table 5. Table 6 summarizes the binding energies (BEs) by UNO CASSCF. From Tables 5 and 6, the following conclusions are drawn: (1) The occupation numbers of dIT-pr. bonding and antibonding NOs are similar between the APUHF and UNO CASSCF. This indicates that the UNO and its occupation number by APUHF (or UHF) is a good trial for the CASSCF calculations. This is in line with our previous conclusion [23] and recent results by Pulay and Hamilton [24]. (2) The BE (25.8kcalmol- 1) for Cr=CHI by UNO CASSCF is close to that (26.3 kcal mol-I) of MCSCF by Mavridis et al. [13]. The BEs by UNO CASSCF are more than 50% of the

S. Yamanaka et al.ll. Mol. Struct, (Theochem) 310 (1994) 185-196

192

Table 5 Occupation numbers (n) and diradical characters (y) for transition metal-methylene cations System

TiCH! VCH! CrCH! MnCH! FeCH! CoCH! NiCH! CuCH!

Method

APUHF CASSCF[5,51 APUHF CASSCF[6,61 APUHF CASSCF[7,71 APUHF CASSCF[8,81 APUHF CASSCF[7,71 APUHF CASSCF[6,61 APUHF CASSCF[5,51 APUHF CASSCF[4,41

a-orbital

,,-orbital

n(a)

n(a")

yea) (%)

n(,,)

n(,,")

y(,,) (%)

1.99995 1.96036 1.99989 1.95592 1.99879 1.92345 1.99616 1.91549 1.99886 1.93304 1.99902 1.93565 1.99977 1.95223 2.00000 1.96430

0.00005 0.03889 0.00011 0.04403 0.00121 0.07688 0.00384 0.08441 0.00114 0.06395 0.00098 0.06351 0.00023 0.04720 0.00000 0.03545

0.005 3.89 0.011 4.40 0.121 7.70 0.384 8.44 0.114 6.40 0.098 6.35 0.023 4.72 0.000 3.55

1.63216 1.67673 1.58865 1.59923 1.55303 1.54549 1.45946 1.38926 1.50601 1.48087 1.52798 1.53947 1.56505 1.60127 2.00000 2.00000

0.36784 0.32340 0.41135 0.40086 0.44697 0.45429 0.54054 0.61072 0.49399 0.51924 0.47202 0.46069 0.43495 0.39912 0.00000 0.00000

36.8 32.3 41.1 40.1 44.7 45.4 54.1 61.1 49.4 51.9 47.2 46.1 43.5 39.9 0.000 0.000

corresponding MCPF values by Bauschlicher et al. [16], whereas they are less than 50% of the experimental values. (3) The UNO CASSCF is effective for computation of the non-dynamical correlation corrections which are responsible for the HF instabilities [23]. The remaining parts are regarded as dynamical correlation corrections, which could be evaluated by the multireference (MR) perturbation and coupled-cluster techniques [23] starting from UNO CAS CI or UNO CASSCF. (4) The APUHF method can reproduce 80% of the BE values by UNO CASSCF except for the case of Cr=CH!. Therefore, the DaDS MO descriptions of the M=CH! bonds in Section 2 are qualitatively correct. This in turn indicates that the UNO CASSCF is useful for confirmation of the DaDS-type bonding pictures.

4. Binding energies between transition metal ions and methylene The UNO CASSCF calculations can include only the so-called non-dynamical part of the electron correlation corrections. Therefore, the BEs of M=CH! by these methods are poor as expected.

Apparently, the dynamical correlation corrections are necessary to obtain reasonable BEs. The larger CAS space compared with UNO CAS is crucial for inclusion of this type of correlation. The expansion of the CAS space for this purpose, however, is not so effective from the viewpoint of computational economy. The second-order perturbation methods Table 6 The transition metal-methylene binding energies by several computational methods using the TZP basis set (BSIII) Methods

Cr

Mn

Fe

Co

APUHF UNOCASCF APUMP2 APUMP4(SDTQ) VCCSD VCC SD(T) QCISD QCISD(T) MCPF" ICACPF" Best estimate" Experiment"

14.3 25.8 41.3 44.3 42.2 46.5 42.6 47.8 39.6 47.9 57±4 52±2

26.2 30.9 58.2 60.3 55.0 58.2 44.3 58.8 45.9 57.0 61±5 69± 3

30.6 37.0 67.6 65.0 67.3 72.1 55.9 73.5 63.4 71.5 74±5 82±5

31.7 38.3 72.9 75.6 74.3 80.0 70.1 82.1 68.3 71.2 79±4 84±5

Heisenberg model

57

71

82

86

a b

Ref. 16. Refs. 1-3.

193

S. Yamanaka et al./J. Mol. Struct, [Theochem} 310 (1994) 185-196

based on CASSCF (CASPT2) [25] and MR CI// CASSCF [13,16] are typical post-HF schemes in the spin-restricted approach. However, MollerPlesset perturbation (MPII), coupled-cluster (CC) or quadratic CI (QCI) methods based on the UHF solution are typical spin-unrestricted approaches. Judging from the diradical character of transition metal-carbon 1r bonds in M=CH! ions, the spinprojections are crucial in the MPn calculations, whereas they are not so serious in the case of the CC or QCI SD(T) approximation under the condition that the diradical character is less than 60% [26]. As shown in other papers [26], APUMP4 and UCC (QCI) SD(T) methods can be utilized in this situation as approximate and convenient procedures for the UNO CASSCF PT2 and UNO CASSCF CC SD(T) methods, respectively: note that the CAS space should be much larger than UNO CAS in order to include a larger part of the correlation correction within the PT2 scheme as shown in the Roos CASPT2 approach [25]. However, the CC approximation is used for this purpose in our approach [23]. The BEs calculated for M=CH! (M = Cr, Mn, Fe, Co) by APUMP4 and UCC (QCI) SD(T) are summarized in Table 6, from which the following conclusions are drawn. (I) The APUMP2 and APUMP4 methods give BE values close to the corresponding values estimated by ICACPF. The MP scheme is reliable for computations of dynamical correlation corrections to UHF if the spin contamination errors are removed adequately [II]. This in turn indicates that the UNO CASSCF PT2 calculations can also provide reasonable BE values for the M=CH! systems. (2) The CC SD and QCI SD methods can also reproduce the ICACPF results [16] at least in the case of the BE values examined here. The spin contamination error in the UHF solution is largely removed because of the use of the exponential excitation operators in the CC scheme. (3) Judging from the BE values obtained by UCC SD, APUMPn (II = 2, 4) slightly overestimates the correlation corrections. (4) The CC and QCI SD(T) methods based on the UHF solution can predict the binding energies by the best theoretical estimations except for the

case of Cr=CH!. The present calculations suggest that the best theoretical value for the species is reduced to 50 ± 3 kcal mol-I. Thus, by the use of the TZP basis set, the UHFbased post-HF methods such as CC SD(T) can reproduce the best theoretical BEs based on the spin-restricted MCPF and ICACPF methods. Numerical data obtained by the former approach wiII be presented elsewhere [27]. However, the UNO analysis indicates that these systems belong to the so-called intermediate correlation regime (the diradical characters are 40%-60%). As is well known, very extensive treatments are crucial in this regime, and therefore the MR CC [23] calculations such as UNO CASSCF CC SD(T) by the use of more flexible basis sets are necessary for conclusive discussions of the BEs of the M=CH! systems.

5. Heisenberg models for M=CH! systems The frontier electrons of the naked transition metal complexes M=Xn+ of our present concern are more or less localized on the M and X atoms as illustrated in Fig. 3. Therefore, the relative stability between the LS and HS states for the systems can also simply be explained by the valence-bond (VB)-like methods such as the Heisenberg (HB) model [28]. The metal-carbon bonding energies are also calculated within the APUMP scheme, where the HB-type energy splittings between the LS, IS and HS states are assumed for the approximate spin projections

H(HB)

= -2 LJpqSp. Sq

(10)

p>q

where J(pq) is the effective exchange integral, and Sp and Sq are orbital spins localized on the M and X atoms, respectively. The spin bond orders (Sp. Sq) [28] are -3/4 and 1/4 for singlet and triplet pairs, respectively. Then the energy difference between the LS and HS states is given by the exchange interactions on the HB model [28] .6.EHs - Ls (H B) =

- 2 [J(OO)

+ J(7i7l") + J(so) + L p>q

Jpq]

(II)

S. Yamanaka CI a/.IJ. Mo/. Struct. [Theocliem} 310 (/994) /85-/96

194

where J(aa) is the direct exchange integral between the da AO of M and the pa AO of CH 2 • while J(7r7r) and J(sa) are those of the ds--pn and sa-da pairs, respectively. The first to third effective exchange integrals in Eq. (11) are proportional to the square of the orbital overlap S and are negative in sign [20]:

J(aa)

= -CS(X)2 (X=da-pa, sa-pa or dr.-pil) (12)

where C is a constant. These are then known as the kinetic exchange (KE) terms. These terms for naked transition metal-carbene systems are regarded as empirical parameters to reproduce experimental BEs:

J(aa)

= -56

J(mr)

= -26 (13)

J(sa) = -15 (kcal mol")

However, the remaining Jpq terms in Eq. (11) are the exchange integrals between the orthogonal orbitals, which are usually known as the potential exchange (PE) terms. The PE interactions are always positive in sign, and their average value K is taken to be 5.5 kcal mol-I. Then the relative stability between the LS and HS states is determined by the balance between the sum of the KE and PE terms as shown in Eq. (11). The LS state is the ground state for transition metal-methylene systems since the KE terms overcome the PE terms. The BE between the transition metal ion and triplet methylene is given by E(cov.) = 3j46.EHS -

LS (H B)

= -~ [J(GeT) +J(mr) +J(sa) + LJPq]

Table 7 Heisenberg models System

Energy formula

Binding energy (BE) Calc. Exp.

~ (J(aa) + J(7:";:-) - 2K) ~(J(aa) +J(;:-;:-) - 4K)

98.4 93.4 80.0 53.8 70.6 83.0 77.8

!

75.2

107 90 74 (J(aa) + J(;:-;:-) - 6K) 57 ~(J(aa) +J(;:-r.) - 8K) 71 ~ (J(sa) + J(aa) + J( .... ) - 9K) 82 ~ (J(sa) + J(aa) + J(7:"r.) - 7K) - U ~ (J(sa) + J(aa) + J(;:-r.) - 5K) - 3U 86 54 ~(J(aa) + J(7:"") - 4K) - 6U (2) (J(sa) + J(aa) + J(7:";:-) - 3K) - 6U 85 47 CuCH! (I) ~ (J(aa) + J( ..r.) - 2K) - IOU 56 (2) !(J(sa) +J(aa) - K)-7U

ScCH! TiCH! VCH! CrCH! MnCH! FeCH! CaCH! NiCH! (I)

!

63.9

taken to be 6 kcal mol-I. Table 7 summarizes the energy formula and the total BEs. There are two different orbital configurations in the case of Ni=CH! as shown in Table 7, leading to the resonance hybrid and the average energy as shown in Table 1. However, the one-electron transferred configuration from M+ to CH 2 is the ground state in the case of Cu=CHt. From Table 7, the calculated BEs for M=CHf are qualitatively in accord with the experimental values. This in turn indicates that the transition metal-methylene systems can be regarded as direct exchange-coupled systems between the transition metal ions M"+ and the triplet monocentric diradical X=CH 2 except for Cu=CH!, for which the bonding scheme is regarded as the direct exchange coupling between Cu(II)2+ and CH2' (case (2) in Table 7).

p >q

( 14) 6. Discussion and concluding remarks However, the on-site Coulombic repulsion U cannot be neglected if the lone pair is formed in the case of rate transition metal ions (M = Fe+ -Cu+) as shown in Table I. Therefore the total energy is given by the sum of E(cov.) and U E(total) = E(cov.)

+ IIIU

(15)

where III denotes the total number of Coulombic repulsions between lone pairs and the average U is

The NO analysis of the UHF solutions for transition metal-methylene ions elucidated the nature of their double bonds. The diradical (DR) character can be estimated by the orbital overlap THO between the split t: orbitals in Eq . (4) [11] and Fig. 3. The ii-DR characters for Cr=CH! are 50, 45, 45 and 40% by the APUHF (BSII), APUHF(BSIlI), UNO CASSCF (BSIlI) and

S. Yamanaka et

sur. Mol. sou«.

GVB [4] methods, respectively. These data show that Cr=CH! is a typical organometallic molecule in the intermediate correlation regime. From the GVB results [28], the 1r-DR character is estimated to be 53% for Ni=CH 2 • As shown previously [11], the spin projections are crucial for UMPn wavefunctions since the significant 1r-orbital splittings entail high-spin contamination. However, the spin projections are not so serious for the UHF CC SD(T) solution [26] under the condition that the DR characters do not exceed 60% as shown in Table 5. This implies that the spin contamination errors become negligible in the CC SDTQ scheme for radicals with larger DR characters. The spin projection is not necessary if the excitation operators are sufficient in the CC approach for the ground-state molecules. Because of the moderate DR characters and spin polarization (SP) effects of the 1r bonds, a localized model such as the HB model can describe the bonding nature of the naked transition metalmethylene ions as shown in section 5. However, the GVB PP model cannot be applicable to the species because of its restriction of the spincoupling scheme, as in the case of other exchange-coupled systems between open-shell species [29]. The APUMPn method has already been applied to the superexchange (indirect exchange) interactions in binuclear transition metal complexes such as MXM (M = Cr(III)3+, Cu(II)2+, X = 0 2- , F-) [30]. It was found that the effective exchange integrals (Jab) in these systems are easily calculated and general tendencies revealed by experiments can be correctly reproduced by the ab initio calculations. The Jab-values of K 2NiF 4-type anti ferromagnetic crystals can also be investigated by the use of cluster models. The Jab-values (about 1000cm- l ) predicted for La2Cu04 on the basis of the APUMPn calculations of Cu(II)02-Cu(II) is found to be consistent with the experimental value (925cm- l ) by the later neutron diffraction technique [31]. The DR characters become less predominant if ligands are introduced into the naked cores. For example, according to the GVB calculations [32], the DR characters are 32% for Clz(O)Cr=CH z, 30% for Clj Ti=CH 2 and 17% forCl z(0)Mo=CH2. These

(Thcochcm) 3/0 (/994) /85-/96

195

results suggest that the 1r-DR character for an 18electron compound (COhNi=CH 2 is calculated to be negligible by the RHF method (THO = 1) [8]. Since the RHF MO result is consistent with several experimental results for the species, its 1r-DR characters may indeed be minor. This further implies that the ,,-DR characters for stable Fischer-type carbene complexes [7,19] could be negligible, and they are also minor or even zero for stable Schrock-type complexes CpzRTa=CRR' [19]. The singlet UHF solution reduces to the closed-shell type. Thus, the ab initio U(R)HF methods are useful and practical enough as a first step for qualitative understanding of the electronic structures of transition metal-carbenes with and without ligands. The methods provide the VB(GVB) and MO(RHF) bonding pictures, respectively, for unstable and stable z-bonds for these species [23]. The diradical characters (y) of 0-30% indicate that the double-bond energies in the metal-carbene complexes can be qualitatively calculated by U(R)HF plus dynamical correlation corrections at the fourth-order Moller-Plesset perturbation (MP4) level. In fact, general trends in the observed bond energies of M=CH! are well reproduced at the APUMP4 level. As shown elsewhere [26], the APUMP4 method is indeed regarded as a convenient alternative to the UNO CASSCF PT2 method. However, the U(R)HF CC SD(T) methods are necessary for semiquantitative calculations of the BEs of labile d1r-p1r bonds of transitionmetal systems, whereas UNO CASSCF CC SD(T)-type calculations [23] are desirable for further quantitative discussions of these systems belonging to the so-called intermediate correlation regime [32]. These are interesting problems for the future.

Acknowledgements This work was supported by a Grant-in Aid for Scientific Research on Priority Areas of Molecular Magnetism (no. 04242101) and by the Asahi Glass Foundation. The UHF-based post-HF calculations were performed by using the GAUSSIAN 9Z program package [33].

196

S. Yamanaka et al.ll. Mol. Struct, (Theochem) 3/0 (1994) 185-196

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