1

Investigation of Spin-Flip Reactions of Nb + CH3CN by

Relativistic Density Functional Theory

Qiang Li1, Yi-Xiang Qiu1, Xian-Yang Chen1, W.H.Eugen Schwarz1,2 and Shu-Guang Wang1,2*

1School of Chemistry and Chemical Engineering, Shanghai Jiao Tong University, 200240 Shanghai, China

2Department of Chemistry, Universität Siegen, 57068 Siegen, Germany

Abstract: In order to explore the details of the reaction mechanisms of Nb atoms with acetonitrile

molecules, the sextet, quartet, and doublet spin-state potential energy surfaces have been investigated.

Density functional theory (DFT) with the relativistic zero-order regular approximation at the

PW91/TZ2P level has been applied. The complicated minimum energy reaction path involves four

transition states (TS), stationary states (1) to (5) and two spin inversions (indicated by ): 6Nb +

NCCH3 → 6Nb η1-NCCH3 (61) → 6TS1/2 4Nb η2-(NC)CH3 (42) → 4TS2/3 → 4NbH η3-(NCCH2)

(43) → 4TS3/4 → CNNbCH3 (44) 2TS4/5 → CN(NbH)CH2 (25). The minimum energy crossing

points were determined with the help of the DFT fractional-occupation-number approach. The first

spin inversion leads from the sextet to an energetically low intermediate quartet (42) with final

insertion of Nb into the C-C bond. The second one from the quartet to the doublet state facilitates the

activation of a C-H bond, lowering the rearrangement-barrier by 44 kJ/mol. The overall reaction is

calculated to be exothermic by about 170-180 kJ/mol. All intermediate and product species were

frequency and NBO analyzed. The species can be rationalized with the help of Lewis type formulas.

Key Words: niobium – acetonitrile – reaction mechanism – potential surface hopping – spin

transitions – minimal energy crossing points - density functional approach – fractional occupation

number approach

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1. Introduction

In the past decades, extensive experimental and theoretical studies1-10 have been carried out on

gas-phase reactions of Transition Metal (TM) atoms and ions, oxide molecules and metal clusters

with alkanes and halomethanes, owing to their fundamental interest as well as potential economic

and environmental significance. As a result, the activation of C-H(X) and C-C bonds by the TM, and

the subsequent H- or X-migration from C to the TM, are now considered as general phenomena11-27.

Recent researches28,29 on various TM-partners indicated H-migration and C-C bond insertion also for

various nitriles.30-35 They act as effective σ and π electron-pair donors for TM coordination, leading

to subsequent rearrangement toward sophisticated products. The studies have provided insight into

the mechanisms and trends of reactivity of TMs interacting with electron lone-pairs or π-bonds of

electron-rich species. Often, geometric multistep rearrangements and the electronic open d-shell of

the TM play decisive roles, initiating the rearrangement of C-H and C-C bonds.

An interesting aspect is the “spin-forbidden” transitions, involving several Potential Energy

Surfaces (PES) with different spins. They have been reviewed under the title Two-State Reactivity36

or Multi-State Reactivit,37-40 meaning that spin inversions between different spin-symmetric states

occur. The so-called spin-accelerated41 reactions become a challenging subject for theoretical

chemists, because of the complexity of two-electron correlation and one-electron relativistic

(Spin-Orbit Coupling - SOC) effects in the TM atoms, and also because of the difficulty of locating

of the Minimum Energy Crossing Point (MECP).

Recent experimental and theoretical works by Cho and Andrews (C+A)29 initiated novel insight

into spin inversion phenomena during the reactions of TM atoms with electron-rich organic species.

An interesting example will here be investigated theoretically in rather detail, namely the following

reaction chain, where we determine the energies and structures of the various intermediates in the 3

possible spin states of sextet, quartet and doublet character, and the respective MECPs:

Nb + N≡C-CH3 → Nb η1-N≡C-CH3 (1) → Nb η2-(N≡C)-CH3 (2) → (1)

Nb η3-(N≡C-CH3) / NbH η3-(N≡C-CH2) (3) → C≡N-Nb-CH3 (4) → CN-NbH=CH2 (5)

At the nonrelativistic level of approximation, two N-atomic PESs of different spin-symmetries

cross at a 3N7 dimensional subspace, the crossing seam. The trajectory that guides the reaction

3

wave-packet will cross in a Low Energy Crossing Region (LECR) around the MECP, the ‘point’

having 3N8 dimensions. With the inclusion of SOC, the MECP changes to an avoided crossing of

saddle-point character lying energetically somewhat below the LECR. A LECR may easily be

reached by the reacting wave packet through thermal activation, and that may outbalance a low

nonadiabatic transition probability (Fig. 1).

Figure 1 Schematic PES (energy of two states S1 and S2 of different electronic symmetries versus reaction

coordinate). (a): Transition with low Boltzmann factor and high adiabatic transition factor. (b) and (c):

Transitions with high Boltzmann factor and low nonadiabatic transition factors.

In the Born-Oppenheimer approximation, one first approaches the 3N-7 dimensional crossing

seam. Afterwards, the spin inversion due to nonadiabatic SOC in the LECR is treated. The algorithm

should account for two conditions: (1) the Franck-Condon condition, i.e. the energies of the two PES

of different symmetry (e.g. 1A and 3B) become the same, E(1A) = E(3B); (2) the classical trajectory

crosses at the lowest energy. The location of the MECP is a practical challenge. Wavefunction

Multi-Configuration (e.g. MCSCF or CASSCF) and Density Functional Theory (DFT) based

procedures have been proposed.42-50 A common approach is calculating the energies and gradients of

the two different electronic states simultaneously for a series of structures of the reaction-complex.

These data are used by an interfaced subroutine, stepwise approaching the MECP.

We have suggested a simple alternative for the search of the MECP in the LECR, applying the

Fractional Orbital Occupation Number (FON) DFT approach.51 In previous works51,52 we have

demonstrated that in various cases the approximate Self Consistent Field (SCF) FON-DFT procedure

could efficiently yield quite good results.

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2. Methodology

Most calculations were performed using the relativistic DFT program from Amsterdam,

ADF2009, initiated by Baerends,53-56 with the spin-unrestricted option. We applied the correlated

local spin density potential of Vosko-Wilk-Nusair (VWN),57 with density gradient corrections for

exchange and correlation of Perdew and Wang (PW91).58 The minimum energy path was followed in

both directions (forward and backward) using the Intrinsic Reaction Coordinate (IRC)59,60 procedure.

It must be admitted that DFT calculations of open shell transition metal shells are only of medium

accuracy, with energy errors of sometimes up to several 10 kJ/mol. We accept this in order to be able

to calculate large numbers of states along the reaction trajectory and find the transition and

surface-crossing points, at a feasible level of computational expense.

The core electrons were calculated by the relativistic Dirac-Slater method61 using accurate

numerical integration, and then transferring the core shells unchanged into the molecules. This is the

so-called all-electron frozen-core approximation, applied to C(1s2), N(1s2) and Nb(1s2-4p6). Standard

Slater-Type-Orbital (STO) basis sets of triple-ζ quality plus two sets of polarization functions

(TZ2P)62 were used for the valence shells of all atoms. Scalar relativistic effects were included using

the Zero Order Regular Approximation (ZORA)63. Analytic harmonic frequencies at the same level

of approximation were determined to characterize the nature of the structures and to evaluate Zero

Point Energy corrections (ZPE).

Weinhold Bond Orbital (NBO)64 and Wiberg Bond Order (WBO)65 analyses were performed

with the help of the Gaussian0366 program, applying a relativistic effective core potential for Nb and

SDD67 and 6-3G*68 basis sets for the metal and the lighter atoms, respectively.

3. Results and Discussion

ZPE-corrected energies of various stationary and transition states on spin-doublet, -quartet and

-sextet PESs along reaction paths starting from Nb(6D) + NCCH3 are shown in Table 1. Table 2

below presents the NBO, WBO and frequency analyses of the two major products CNNbCH3 and

CN(NbH)CH2 in the three spin states. Table 3 displays CN-bond parameters to better understand the

contribution of the CN group during the multi-step process. The reaction path energies are plotted in

Fig. 2. Fig. 3 provides the most important geometric parameters at the stationary and transition points.

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Some of the species calculated here were also calculated by C+A29, using small-core

pseudopotentials and a TZ+Pol GTO basis, though without polarization functions on the Nb. The

B3LYP energies of C+A from their Fig. 8 (after correcting for a data interchangd) scatter around the

present PW91 values with a σ of about ±25 kJ/mol (not much for open d-shell DFT), see Table 1.

C+A’s and our PW91 vibration frequencies also agree reasonably well with each other as well as

with the few measured values, see Table 2.

The ground state of the reacting Nb atom is the high-spin sextet 6D1/2 (4d45s1). On the

background of their experimental observations and theoretical B3LYP/BPW91 computations, C+A29

assumed that the major products CNNbCH3 and CN(NbH)CH2 show up in ground doublet and

quartet states, respectively, and proposed a mechanism for the Nb + acetonitrile reaction. Assuming

pure spin states (i.e. averaged out SOC), this reaction path has here been basically reproduced but

amended with many additional details, see Figs. 2 and 3.

TABLE 1 Calculated Energies a) (in kJ/mol) of Intermediates, Transition States and Products relative to the

Ground State Reactants Nb(6D) + CH3CN. Values in parentheses are from Fig.8 of C+A.29.

Species Sextet Quartet Doublet

Nb:

Nb+NCCH3

d4s 6D

- 0 -

d3s2 4F

+17.8 b)

d3s2 2G

+101.4 b)

1 -145.2 (~ -115) -86.6 (~ -75) -5.3 (~ -35)

TS1/2 -76.4 -62.1 -4.5

2 -103.2 (~ -80) c) -178.9 (~ -145) -143.9 (~ -120)

TS2/3 +71.6 -49.6 -55.7

3 +60.9 -175.3 -141.3

TS3/4 +61.4 -66.2 -21.9

4 +38.2 (~ +40) c) -182.3 (~ -200) -136.7 (~ -155)

TS4/5 +205.2 -55.5 -99.8

5 +201.5 (+ large) -59.1(~ -95) -169.8 (~ -175) a) ZPE-Corrected, from SO averaged ZORA-PW91-TZ2P/STO DFT. b) The relative atomic average term energies are from spectroscopy. c) The two entries had obviously been interchanged by C+A in their Figure, according to

our recalculation according to their procedure.

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Figure 2 Energies (ΔE in kJ/mol) on the reaction path Nb+NCCH3 → CN(NbH)CH3 in sextet, quartet, and

doublet spin-states.

3.1 Spin-conserving Reaction Mechanism of the Nb(2D) Doublet State

In the high-energy low-spin doublet term 2G (and in the energetically adjacent 2D and 2P) of the

d3s2 configuration, the Nb atom has several holes in the d-shell and can efficiently act as an

electron-pair acceptor for 2Nb←NCCH3 dative bonding, forming a common low-spin coordination

complex. The addition of the metal atom does not break the symmetry of acetronitrile, 2Nb

η1-NCCH3 (21) keeping the C3v structure. The relative complex stabilization energy of 21 is more

than -100 kJ/mol, C+A found a similar value. However, 2NbNCCH3 would immediately convert over

a negligible barrier 2TS1/2, the Nb moving out of the molecular axis to above the nitrile triple bond,

forming the nitrile η2-π-complex 22. Remarkably, there is an additional stabilization energy of nearly

-140 kJ/mol, making 22 more than -240 kJ/mol lower than the doublet educts.

The next step is the insertion of Nb into the NCCH3 bond. C+A assumed, in consideration of

the energy content of the Nb+NCCH3 reaction complex, a direct movement of M (M = Zr, V, Nb, Ta)

from the η2-π-complex 2 into the C-C bond forming species 4. Our IRC investigation revealed

complicated PESs and rearrangement processes for this movement in all three spin states. First, from

intermediate 22, the reaction proceeds over a significant transition state barrrier 2TS2/3, the metal

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atom approaching the methyl group and capturing the H1 atom. Then intermediate 23 is formed with

cluster structure 2Nb η3-(N-C2-C1H2), see Fig. 3. The Nb-C1 bond distance becomes as short as 2.22

Å, which is just the sum of the atomic single bond radii of Nb and C69. The IRC calculation confirms

that transition state 2TS2/3 is connected to intermediate 23 in the forward direction and to

intermediate 22 in the backward direction.

Then, from 23, Nb continues moving toward C1 and eventually breaks the C1-C2 bond.

Remarkably, the Nb-N bond remains preserved, i.e. the nitrile group is turned around, while a new

carbon-metal H3C-Nb bond is formed and H1 returns to the methylene. The nitrilic N-C2 triple bond

remains basically intact during all these rearrangements. The barrier of TS2/3 (88 kJ/mol) is smaller

than the barrier of TS3/4 (119 kJ/mol). The latter one is then expected to be the rate-determining step

on the whole doublet reaction path. Eventually, the reaction may proceed to produce

CH2=(NbH)NC (25) via transition state 2TS4/5 with a barrier of 37 kJ/mol and final release of 33

kJ/mol energy The whole reaction on this path is exothermic by about 270 kJ/mol.

3.2 Spin-conserving Reaction Mechanism of the Nb(4F) Quartet State

According to Fig. 2, the quartet and doublet reaction-paths are somewhat similar. However, the

quartet requests a slight activation to surmount the barrier (25 kJ/mol) from linear 41 to the η2-π-

complex 42. Intermediates 42, 43, 44 are very similar in energy (as were 22, 23, 24) and they are the

lowest among the three spin states. The transition barriers at 4TS2/3 and 4TS3/4 are slightly higher

than on the doublet path, 129 and 109 kJ/mol, respectively. Formation of 45 with H1 migration from

C1 to Nb, however, requires significantly more energy than in the doublet case and would end up at

111 kJ/mol above the 25. 45 is expected also as kinetically unstable, converting practically without

barrier into 44. The quartet reaction from 4Nb + H3CN to final species 44 is exothermic, by about 170

kJ/mol.

3.3 Spin-conserving Reaction Mechanism of the Nb(6D) Sextet State

The ground state of the reactant Nb atom is 6D1/2 (4d45s1). The SOC-averaged 4F and 2G terms

lie ~18 and ~101 kJ/mol, respectively, above the 6D. Analogous to the quartet and doublet cases, 6Nb

8

9

Figure 3 Optimized geometries of the intermediates, the transition states and the products of the reaction of

Nb+NCCH3 in the (a) sextet, (b) quartet, and (c) doublet spin states (distances in Å, angles in o).

may at first be linearly attached to the acetonitril in 61. The η2-π-complex 62 is formed, however, this

time with a noticeable transition barrier of 69 kJ/mol. In contrast to the quartet and doublet paths, the

Nb insertion into the C1-C2 bond from 62 to 64 can take place only upon energetic activation through

structurally unstable intermediate 63 and a crowded transition states, which both differ from the

structures on the quartet and doublet PESs. 6Nb does not well support Nb-CH3 nor Nb=CH2 bonds,

the weak interaction between C1 and Nb being ~2.5 Å long. An enormous activation energy of 175

kJ/mol over the 6TS4/5 barrier would be needed for the transition from 64 to 65. Unlike the final flat

CH2=Nb(H)NC structures on the quartet and doublet PESs, 65 shows up as a bent structure

negligibly lower behind the high barrier. Thus, the later steps on the sextet PES seem improbable, the

sextet reaction getting stuck at 61 or 62.

3.4 Spin Crossing and the Probable Overall Reaction Path

The lowest terms of Nb atoms are the adjacent 6D and 4F ones. The ground states of the educts

and of the first (linear) η1-σ-adduct (1) are sextets. The ground states of the η2-π-complex (2), of the

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NbH-η3-complex (3) and of the H3C-Nb-NC structure (4) are quartets. The ground state of

H2C-NbH-NC (5) is a doublet. The four different structures (2) to (5) are rather similar in energy,

lying between -170 (25) and -182 (44) kJ/mol, which is not significantly different at the accuracy

level of the present methodology. A low energy path would go on these three PESs, and several spin

crossings might occur. The 3d-, 4d- and 5d-TM mediated reactions are well known to occur on more

than one adiabatic PES.50,70-75

The reaction of laser ablated Nb atoms will start from 6Nb with the formation of labile

σ-complex 61 on the sextet PES, and possibly also from 4Nb with the formation of more labile

σ-complex 41 on the quartet PES. The sextet reaction trajectory crosses the quartet (and also the

doublet) surface at Projected Crossing Point 1 (PCP1 in Fig.2) between 6TS1/2 and intermediate 42.

Thus, the reaction should then proceed further on the quartet surface form 42 via 4TS2/3 to 43, then

via 4TS3/4 to 44, and possibly partially also on the doublet surface via 2TS2/3, 23 and 2TS3/4 to 24.

Eventually, the reaction trajectory might go via 2TS4/5 to 25, possibly after a jump from the quartet

to the doublet PES at PCP2. There may occur jumps between the quartet and doublet PESs also

earlier between 2 and TS2/3, between TS2/3 and 3, between 3 and TS3/4, and between TS3/4 and 4.

However, there the structures of the quartet and doublet states are more different, which may cause

significant spin-flip barriers. So, the reaction may stop at η2-complex 42, η3-complex 43,

doubly-coordinated Nb-compound 44 or, provided 25 is low enough, at a triply-coordinated

Nb-complex. A possible low-energy pathway through PCPs to this very end is

Nb(6D) + NCCH3 → Nb η1-NCCH3 (61) → 6TS1/2 → PCP1 →

Nb η2- (NC)CH3 (42) → 4TS2/3 → NbH) η3-( (NCCH2) (

43) → (2)

4TS3/4 → CNNbCH3 (44) → PCP2 → 2TS4/5 → CN(NbH)CH2 (

25).

If the reaction starts on the sextet PES and ends on the doublet PES at 25, the overall reaction would

be exothermic by around 170 kJ/mol (C+A: ~175 kJ/mol). If the reaction would already stop on the

quartet PES, the reaction would be exothermic by ~180 kJ/mol, which is not distinguishable from

170 kJ/mol within the error bars. C+A found ~200 kJ/mol for 44, with possibly larger error due to

basis deficiency and the known problems of B3LYP with heavy atoms.

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3.5 Crossing Points between the PESs of Different Spin Multiplicities

The electronic ground states of many molecules around their equilibrium geometry can be well

approximated by a single determinant wavefunction without significant near-degeneracy

configuration mixing. Then, DFT approaches are known to work reasonably well. In such ground

states, the electrons occupy the orbitals with lowest DFT-orbital-energies. But in some special cases

such as bond breaking and formation at larger distances, bond rearrangement at transition points,

near-degenerate ground states, or low-lying excited states, non-dynamic correlation effects become

important.

LECR1. The first crossing is near 6TS1/2 and 4TS1/2. In the sextet state, among the 6 higher

orbitals of Nb-3d-5s type, the lower five α-spin-orbitals (9Aα1 – 13Aα1) are (singly) occupied. 14Aα0

and the respective six β0-spin-orbitals are higher in energy and empty, see Fig. 4. The quartet is

obtained by transferring the highest α-electron from HOMO 13Aα to LUMO 9Aβ. Thereby the

13Aα0 orbital energy raises and the 9Aβ1 orbital energy falls below it, becoming the HOMO of the

quartet. I.e. both electronic configurations of different spin-symmetry obey the “Aufbau principle”.

Figure 4 Spin-orbital level schemes of 6TS1/2 (left) and 4TS1/2 (right) for the 5 HOMOs and 3 LUMOs,

spin-orbital energies ε in eV, from ZORA-PW91-DFT-UHF calculations.

The question is now, how to handle the transition between them. SOC would mix the two

configurations. Only SOC multi-configuration methods such as SOC-CASSCF or SOC-MR-CI are

suitable for rigorously locating the transition region. On the other hand, any quantum chemical

procedure has its accuracy range, and further, the concept of a classical trajectory over a transition

point is an approximate model anyhow, the reacting wave packet sliding around the minimal energy

12

path. Here, we apply a simple, easy and efficient way of approaching the LECR, namely by the

FON-DFT method.51

As noted above, the first spin-flip may occur between 6TS1/2 and 42, i.e. when Nb moves from

the nitrogen lone-pair end to above the -C≡N π-bond. In the quartet state, empty and doubly

occupied Nd-4d AOs (instead of half-filled AOs in the sextet) can better act as acceptor and

back-donor towards the nitrile group. Form 62 to 42, Nb moves 0.3Å nearer to the triple bond, which

expands by 0.06 Å. A CN-π-pair partially moves into an empty Nb-4d5s-σ orbital, while the αβ-pair

of 4Nb interacts in a back-donating manner with the nitrile π* orbital. To describe the transition path

in two dimensions x,y, we chose a reaction coordinate x, which varies significantly from the initial to

the final state. As Fig. 3 shows, the Nb-N-C2 angle is an appropriate choice for x. The second

orthogonal dimension y corresponds to the change of the respective occupation numbers of the two

active spin-orbitals, 13Aα1-n and 9Aβn. Simultaneously with the n-variation, the remaining 3N-7

structural parameters of the N=7-atomic system are optimized. Near the crossing point CP1, the

structural variation is dominated by the distance changes of Nb···CN and C≡N. Fig. 5(a, top) is a

projection onto the reaction coordinate, showing the “projected crossing point” PCP1 at ∠Nb-N-C2

112.1° with E(6A)= E(4A). The two electronic configurations have the same ∠Nb-N-C2 angle, but

different 3N-7 dimensional partial structures. The location of the PCP is the first step of our

procedure.

In the second computational step, the MECP is approximately located in the second dimension

(∠Nb-N-C2 angle fixed) with the help of the FON procedure. All other geometric parameters are

optimized (‘partial structure optimization’) together with the fractional occupation number n of

orbitals 13A(α)n and 9A(β)1-n. When n decreases from 1 to 0, orbital 13A(α)n shifts up and 9A(β)1-n

goes down. For n = 0.487, the two orbital energies become equal (Fig. 5a, bottom). This geometry is

near to the MECP1 (Fig. 6, left).

The last step is to check whether a second search cycle is necessary, which however is rarely the

case. To this aim the two single-configuration energies of 6A and 4A at the LECR-point are compared.

The difference was only 0.3 kJ/mol. So, at this point, the Franck Condon principle is satisfied for the

two different spin-multiplicities having the same geometry and energy. Both states also satisfy the

“Aufbau principle”, no unoccupied orbital is below an occupied orbital. Both states are the lowest of

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their spin symmetry, fulfilling Janak’s theorem76.

Figure 5 Left (a) 6/4TS1/2 from Nbη1-NCCH3(6/41) → Nbη2-NC-CH3(

6/42). Right (b) CN-Nb-CH3(4/24)

→ 4/2TS4/5 to CN-(NbH)=CH2(4/25). Top: Optimized energy curves E of pure spin states along the reaction

coordinates ∠Nb-N-C2 and ∠H1-Nb-C1, respectively. Middle: Ensemble FON energy curve E versus

occupation number n and simultaneously optimized partial structure. Bottom: Orbital energies εi for

optimized partial structure when varying the FON n: for 13A(α)n and 9A(β)1-n of mixed ensemble (1-n)(4A) +

n(6A), and respectively, for 12A(α)n and 10A(β)1-n of mixed ensemble (1-n)(2A) + n(4A).

Another common approximate procedure43 for locating the crossing point has also been

examined, namely optimizing the 6A state along the reaction path and determining for these

structures the energies of the 4A state, or vice versa, for interchanged 6A and 4A states. However, no

crossing was found along these two reaction paths, with remaining unacceptable energy difference

between the two states51.

LECR2. For the quartet-doublet transition between 44;24 and 4TS4/5;2TS4/5, we applied the

∠H1NbC1 angle as the reaction coordinate, see Fig. 3. The two electronic states 4A and 2A take the

same energy at the PCP2 at angle ∠H1NbC1 = 50.05°. After FON optimization (Fig. 5b, bottom),

we got the structure of the MECP2 (Fig. 6, right). There is a 1 kJ/mol energy difference between the

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pure 4A and 2A states, which is still acceptable.

Figure 6 Structural parameters of the two MECPs (bond lengths in Å, bond angles in o)

3.6 Molecular Structure and Bonding

Concerning the stationary states calculated by C+A, their B3LYP structures agree with ours. We

will here focus on the major products CNNbCH3 (4) and CN(NbH)CH2 (5). The structural and

energetic details of the six species 2.4.64,5 (Table 2, Fig. 3) can be partly rationalized with the help of

the Lewis formulas displayed in Fig. 7. The orbital picture turns out to be rather involved, for several

reasons. The Nb atom has 6 orbitals (12 spin-orbitals) in its 4d-5s valence shell, for only 5 electrons.

The respective flexibility of the valence shell enables the approximate description of

multi-configuration states by symmetry-broken determinants. The correlated spin-singlet of a doubly

occupied orbital may then be represented by two different spin-orbitals φ1α and φ2β, which also may

contribute to atomic spin-densities of the spin-polarized open-shell molecule. And while the lowest

spin-triplet [φ1α1, φ2α

1] of a pair-bonded hydrogen molecule [φ1α1, φ1β

1] is antibonding, the triplet in

the present case may, and is, still be partially bonding.

We begin the discussions of the quartets with product ground state 44, and subsequently of the

doublets with product ground state 25. Two electrons of the five ones on Nb are used in 44 to form

C1-Nb and N-Nb single bonds at 129o. Three electrons are left over, forming a (4d5s)3 spin-quartet

(Fig. 7 middle left). The spin-coupling of two electrons to form a statically correlated pair and the

spin-doublet 24 (Fig. 7, top left) costs energy according to Hund’s rules, here ~0.5 eV, and the

Nb-spin density decreases from ~3 to ~ 1. The transition 44 → 64 requires the breaking of a weak pair,

the NbC1 single-bond, raising the energy by ~2.3 eV, expanding the internuclear distance by 38 pm

and opening the C1-Nb-N angle by 32o, though not completely destroying the attractive C1-Nb

15

interaction. The spin densities become 4 at Nb, and nearly 1 at C1. Remarkably, 0.2 units of

spin-density are transferred to the other end of the molecule, see Fig. 7, left bottom. Numerical

property values of the states are displayed in Table 2.

Figure 7. Lewis formulas of the three spin-species of products 4 and 5 (only fractional spin densities on atoms

are indicated explicitly).

In 25, two electrons from Nb are used for the σπ-double bond to CH2, and one electron each for

the single bonds to H1 and NC. Remarkably, the remaining electron gives rise, according to our DFT

calculation, to ~1.2α spin density on Nb and ~0.2β spin-density on H1. Transition 25 → 45 breaks the

NbC1 π-bond pair, raises the energy by 1.15 eV and expands the internuclear distance by 23 pm,

the spin density increasing by ~1.1 at Nb and by ~0.9 at C1 (Table 2; Fig. 7 right). Another pair

breaking for 45 → 65 affects both the Nb-C1 and Nb-H1 bonds, expanding them by 28 and 18 pm.

The energy is raised by 2.7 eV, the angle between the two weakened bonds widens by nearly 30o.

The spin-density increases are complex, 0.8 on Nb, 0.6 on H1, 0.4 on C1, and 0.2 C2.

The bond lengths (Fig. 3) and bond-types (Fig. 7) correlate well with the Wiberg bond orders

(WBO in Table 2)65. The Nb=C1 double bond in 25 has WBO=1.95, the NbC1 single bond in 45 has

WBO=0.95, and the bond-weakening in the sextet corresponds to WBO=0.74. Compared to

Pyykkö’s bond radii69, the lengths of the NbCH3 single bond in 14 (~209 pm) shows δR = 13 pm

short, while (~218 pm, δR = 4) in 34, 35 agree better, those of the NbH1 single bonds (~187 pm) in

15 and 35 are 8 pm short, and of the ZrC1 double bond (194 pm, δR = 0 pm) in 15 agree well. Also

the Nb-C1 stretching frequencies in Table 2 of 798 cm-1 for the doublet (observed at 788.8 cm-1;

16

C+A/PW91 802 cm-1), for the quartet of 583 cm-1 and for the sextet of 472 cm-1 show the

corresponding trends. The Nb-C1 frequency of 24 and 44 of ~600 cm-1 goes down to ~230 cm-1 for

the 64, with WBOs of ~0.8 and~ 0.2, and bond lengths of ~2.1 and ~2.5 Å, respectively. The dative

|C≡N→Nb bonds have a rather constant length of ~204 pm in all these complexes.

TABLE 2. NBO, WBO and Frequency (ν) Data of the Major Products CNNbCH3 (4) and CN(NbH)CH2 (5) in

the 3 Spin States

Property H3C Nb NC H2C NbH NC

24 44 64 25 45 65

q(C1) a -1.13 -1.15 -0.69 -0.82 -0.64 -0.53

q(H1) a 0.28 0.27 0.22 -0.29 -0.25 -0.08

q(H2&H3) a 0.25 0.25 0.24 0.22&0.24 0.21 0.20

q(CH3/CH2) a CH3:-0.35 CH3:-0.38 CH3:0.01 CH2:-0.36 CH2:-0.22 CH2:-0.13

q(C2) a 0.26 0.25 0.25 0.24 0.27 0.28

q(N) a -0.84 -0.85 -0.81 -0.85 -0.84 -0.77

q(CN) a -0.58 -0.60 -0.56 -0.61 -0.57 -0.49

q(Nb) a 0.93 0.98 0.55 1.25 1.04 0.70

C1 b 2s1.25 2p3.87 2s1.25 2p3.89 2s1.25 2p3.43 2s1.28 2p3.52 2s1.36 2p3.27 2s1.34 2p3.17

Nb b 4d3.25 5s0.80 4d3.36 5s0.63 4d3.76 5s0.67 4d3.46 5s0.29 4d3.35 5s0.59 4d3.67 5s0.62

v(Nb-CH2,3) c 612 578 230{799}g 798/(789) 583 472

v(Nb-NC) c 470 475 400 482 459 437

v(Nb-H) c 1702/(1708) 1752 1008 e

v(C2-N) c 1956 1977/(2046) 1965 1998/(2031) 1987 1942

WBONb-C1 d 0.93 0.89 0.21 1.95 0.95 0.74

WBONb-N d 0.73 0.69 0.59 0.53 0.70 0.67

WBONb-H1 d 0.85 0.90 0.45

ρα(Nb) f 1.08 3.06 4.01 h 1.18 2.32 3.13 h

ρα(C1) f -0.02 -0.10 0.84 -0.12 0.79 1.16

ρα(H1) f -0.03 -0.08 0.50 a Weinhold charge. Values of free H3CCN: q(N): -0.32, q(C2): 0.27, q(CN): -0.03, q(CH3): 0.03. b Weinhold electron configuration. c Frequency in cm-1, experimental value of C+A in parentheses 29. d Wiberg bond order. e A low intensity vibration with Nb-H contribution appears at 1219. f Surplus of atomic α-spin density. g The H3C umbrella vibration in braces has also some C-Nb component and much more intensity. h A missing part of 0.19 / 0.21 spin density is on the terminal C2 of the cyano group in 64 / 65, respectively.

17

18

The CN moiety changes in a characteristic way in these arrangements. In the educt R-C≡N|, the

experimental77 C-N and C-CN distances are 116 and 146 pm, respectively (here calculated as 116

and 145 pm), with WBO = 2.90 and 1.09. Upon N-Nb σ-bonding, of the dative η1 or covalent type in

structures 1, 4 or 5, RCN increases by 2-3 pm, the WBO decreases by half a unit (Table 3) and the

C-N frequency is reduced by ~200 cm (vCN of free acetonitrile: calc. 2282 cm, exp.78 2267 cm).

However, in the π-complexes of η2 and η3 type, 2 and 3, and in the transition states TS2/3 and TS3/4,

the Nb atom interacts more strongly with C≡N (RCN widens by 5 up to 20 pm, WBO goes down by

up to more than one unit in some structures) and also with RC (RCC is already expanded by 2-5 pm

in 2). WBOs and bond lengths correlated nicely, with ΔRCN/pm ≈ 10·ΔWBOCN. The effects increase

from the sextet to the quartet to the doublet, when the Nb atom becomes more suited for electron-pair

acceptation and charge back-donation. I.e. interaction of Nb with the soft CN moiety stabilizes its

low and intermediate spin states, and vice versa, affecting the nitrile makes the NCCH3 bond prone

to insertion.

TABLE 3. Changes of Bond Length ΔRCN (in pm) and Wiberg Bond Orders (ΔWBO in parentheses) of the

CN Group in Intermediates and Products in the 3 Spin States. *

Sextet Quartet Doublet

acetonitrile ΔRCN = - 0 - , RCN = 115.9 pm (ΔWBO = - 0 -, WBO = 2.90)

1 +2.3 (0.52) +2.2 (0.52) +2.2 (0.58)

2 +4.8 (0.56) +12.7 (1.17) +12.9 (1.18)

TS2/3 +3.9 (.45) +8.9 (.83) +21.5 (1.76)

3 +4.3 (0.50) +5.0 (0.67) +7.0 (0.73)

TS3/4 +4.0 (.51) +7.1 (.69) +7.8 (.77)

4 +2.8 (0.46) +3.2 (0.53) +3.3 (0.56)

5 +2.8 (0.49) +3.1 (0.54) +3.0 (0.51)

* From PW91/6-3G* DFT

4. Conclusions

The path for the reaction of Nb atoms with acetonitrile molecules has been investigated with the

help of quasi-relativistic DFT, exploring the sextet, quartet and doublet PESs. The following results

were obtained.

19

1. Starting with the sextet ground state of the Nb atom and the respective η1-σ-complex of

acetonitrile, η2,3-π-complexes are formed at first, crossing from the sextet to the quartet PES near the

sextet and quartet saddle-points. A single η1-σ-interaction with the nitrile keeps the high-spin sextet

ground state of the Nb atom, while a single η2-π-interaction already stabilizes the intermediate

quartet state. The NCCH3 bond becomes activated through the influence of the Nb atom on the

C≡N bond in the π-complexes.

2. A second spin crossing point was located near the saddle-points on the quartet and doublet

PESs between the H3CNbNC and H2C=NbHNC structures. The minimum energy pathway can

be described as follows (underlined species were already experimentally identified by C+A29):

6(4)Nb+NCCH3 → Nb η1-NCCH3 (6(4)1) → 6(4)TS1/2 →

Nb η2-NCCH3 (4(2)2) → 4(2)TS2/3 → (NbH) η3-NCCH2 (

4(2)3) → 4(2)TS3/4 → (3)

CNNbCH3 (4(2)4) —ε→ 2TS4/5 → CNNbH=C H2 (

25) .

I.e. the second spin-crossing to the low-spin doublet state occurs, when a methyl C-H bond becomes

activated by the Nb. This spin inversion leads to a decrease of the reductive elimination barrier

height by about 45 kJ/mol to 83 kJ/mol.

3. The reaction, starting on the sextet PES, may end on the quartet PES at species 42, 43 or 44,

or on the doublet PES at species 25, depending on whether sufficient energy is available to surmount

the barriers around 100 20 kJ/mol. The most stable intermediates and products, all at similar

energies ~175 kJ/mol below the reactants, are also the ones that can be well represented by Lewis

formulas.

4. All local minima structures of the three spin states were investigated by frequency analyses.

Bond lengths, frequencies and bond orders are consistent with the Lewis structural models. The Nb

atom carries an effective Weinhold charge of about +1 in all lower energy species. The Wiberg bond

orders indicate significant σ donation and π* gain (back-donation) of the CN triple bond to/from the

Nb atom in the quartet and doublet states of the η2-π-complex. The Nb atom activates C-H and C-C

bonds in intermediate and low spin states in combination with the adjacent nitrile group.

20

5. When starting with a nitrile, which interacts with the Nb atom through its nitrogen atom, the

Nb-N η1-σ-interaction survives all rearrangements. This holds for the intermediates 2, TS2/3, 3 and

TS3/4, although the Nb-C2 distance may become as short as a Nb-C single bond and sometimes

even shorter than the Nb-N bond. The shapes of the PESs do not support the replacement of a Nb-N

bond by a Nb-C2 bond, although for instance the isomer H3C-Nb-CN is a little more stable (by about

12 kJ/mol) than 4 (H3C-Nb-NC). We also note that the equilibrium distances of the Nb-N bonds are

always near 2 Å, while the Nb-C distances show a much larger variation between 1.9 and 2.5 Å.

6. There remain some open questions. The 25 state is experimentally obtained from the 44 state

upon energetic activation. Therefore, 25 should be somewhat lower in energy than 44, while our DFT

results placed it 12.5 (C+A: 25) kJ/mol above 44. Also concerning the other states, more accurate

energies would be welcome, obtainable for instance from large basis CAS-PT2. Such MC-SCF

results would also allow for a reliable description of the open d-shell. In particular the unusual

bonding situation in the high-spin states (such as 64, 45, 65) and the spin-density distributions could

be elucidated.

Acknowledgement. We acknowledge financial support by the National Nature Science Foundation

of China (No. 20973109).

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24

24(Nb) 44(Nb) 64(C)

25(Nb) 45(C1) 65(NbH)

Figure 8 Typical upper occupied canonical spin-orbital envelops (contour value 0.03 au) from PW91-DFT.

24(Nb): Spin-uncoupled occupied α-orbital of Nb-dxy character with slight N-C2-pπ contamination. 44(Nb):

Nb-4d5s-type spin-uncoupled occupied α-orbital. 64(C1): C1-pσ - C2-pπ type spin-uncoupled occupied α-

orbital. 25(Nb): Spin-uncoupled occupied orbital of Nb-dxy character with slight N-C2-pπ contamination.

45(C1): Spin-uncoupled occupied α-orbital on H2C1 with Nb and H1 contamination. 65(H): Nb-4d5s-type

spin-uncoupled occupied orbital on and H.