Computational Studies of the Electronic Spectra of Transition-Metal-Containing

Molecules

James T. Muckerman, Zhong Wang, Trevor J. Sears

Chemistry Department, Brookhaven National Laboratory, Upton, NY 11973-5000 USA

and

Hua Hou

Chemistry Department, Wuhan University, Wuhan, P.R. China

62nd Ohio State University International Symposium on Molecular SpectroscopyJune 18-22, 2007

Outline

• Lingering issues regarding our previous calculations on FeH and VH

• Diabolical features of the low-lying electronic states of TiC2

• The electronic ground state of the titanium metcar, Ti8C12

Focus on the questions that make the treatment of these problems difficult

4

X 4

A 4

a 6

b 6 C 4

c 6

E 4

F 4

B 4

D 4

X 5

5

5

5

5

3

3

3

3

Lingering issues regarding our previous calculations on FeH and VH

Lingering issues regarding our previous calculations on FeH and VH

6 1 7 3

8 4 9

10 11 5

Lingering Issues Regarding our Previous Calculations on FeH and VHFocus on the C 4 state of FeH, the B 5+ state of VH, and dipole moments

We believed that our previous results for the C 4 state of FeH (C(0)~0.89) and the B 5+ state of VH (C(0)~0.96) could be improved with a better choice of reference function such as we did for the B 4 and D 4+ states of FeH.

It would probably be better not to use the same orbitals for the 4 and 4 states of FeH and the 5± and 5 states of VH, but rather to perform separate CASSCF calculations with a large weight assigned to the problematic states, followed by MRCI calculations in which the ground and target state roots are computed but only the target state root is retained.

Our Te value for the C 4 state of FeH (3791 cm-1) is considerably smaller than other calculated values (4759 to 6416 cm-1), but (perhaps fortuitously) close to the experimental value (3175 cm-1). [C. Wilson et al., J. Chem. Phys. (2001)]

Our calculated vibrational frequencies are in general agreement with available experimental data and most other calculated values except for the C 4 state of FeH for which our value is considerably smaller.

More Lingering Issues Regarding FeH and VH…

Our values of the dipole moments of the various FeH states were computed as the expectation value of the dipole operator with the MRCI wavefunction, and tend to be smaller (by 0.7 to 1.0 D) than the available experimental values and most other theoretical values.

They do, however, reflect the experimentally observed [T. C. Steimle et al., J. Chem. Phys. (2006)] difference between the values for the X 4 and the F 4 states of FeH that, according to our analysis, arises from the F state originating primarily from the 17 configuration of the Fe atom while the X state originates primarily for the 26 configuration.

The dependence of the computed dipole moments on internuclear distance shows the F state having a smaller magnitude at the equilibrium internuclear distance but a much larger derivative than the X state.

These results contrasts with those of Tanaka et al. [K. Tanaka et al., J. Chem. Phys. (2001)] that predict similar values and derivatives of the dipole moments of the F and X states.

Still More Lingering Issues Regarding FeH and VH…

With the exception of the B 5+ state, there is remarkably little difference between the relative energetics of the MRCI and MRCI+Q results of VH.

The B 5+ state of VH with principal configuration 22 is stabilized relative to the X state by 0.126 eV by the Davidson correction.

This does not seem to be an artifact of a small C(0), which hovers around 0.96, but results from a somewhat larger correlation energy (ca. -0.08 compared to ca. -0.05 Hartree) than the other states.

We also could not follow the B 5+ state to separations less than 1.65 Å because this second root of 5A1 symmetry converges to a higher-lying state.

Our calculated C 5 state, the second root of 5A2 symmetry with principal configurations 2 and 2, does not behave in this manner.

F 4

A 4X 4

b 6

c 6

C 4

a 6

D 4

B 4

E 4

F 4

A 4X 4

b 6

c 6

C 4

a 6

D 4

B 4

E 4

Change in FeH C 4 state MRCI energy by state average re-weighting

80% A 4:10% C 4:10% E 4 30% A 4:70% C 4:0% E 4

C(0)~0.96C(0)~0.89

X 4

A 4

a 6

b 6 C 4

c 6

E 4

F 4

B 4

D 4

X 4

A 4

a 6

b 6 C 4

c 6

E 4

F 4

B 4

D 4

Change in FeH C 4 state MRCI+Q energy by state average re-weighting

80% A 4:10% C 4:10% E 4 30% A 4:70% C 4:0% E 4

X 5

5

5

55

3

3

3

3

X 5

5

5

5

5

3

3

3

3

Change in VH B 5+ state MRCI energy by state average re-weighting

80% X 5:10% B 5+:10% C 5 10% X 5:85% B 5+:5% C 5

C(0)~0.98C(0)~0.96

X 5

5

5

5

5

3

3

3

3

X 5

5

5

5

5

3

3

3

3

Change in VH B 5+ state MRCI+Q energy by state average re-weighting

80% X 5:10% B 5+:10% C 5 10% X 5:85% B 5+:5% C 5

X 5

A 5

D 5

B 5

a 3

d 3

c 3

b 3 C 5

80% X 5:10% B 5+:10% C 5 10% X 5:85% B 5+:5% C 5

Change in VH B 5+ state MRCI dipole moment by state average re-weighting

States X 5 A 5 B 5+ C 5 D 5 Re (Å) 1.7111 1.7386 1.8075 1.8480 1.8377 Expectation value 2.5942 2.5168 4.7712 1.3635 1.2632 Field calculation 2.6518 2.9093 8.8387 1.2068 1.1662

Dipole Moments (in Debye) of VH Quintet States at their Energy Minima

States X 4 F 4 Re (Å) 1.5684 1.7176 1.5684 1.7176 Expectation value 1.8982 1.9427 0.0106 0.3279 Field calculation 2.6497 2.7188 0.0781 0.3916 Observed a 2.61 1.27

Dipole Moments (in Debye) of FeH 4 States at their Energy Minima

Dipole Moments by Expectation Value vs. Applied Electric Field Calculations

a C. Wilson et al., J. Chem. Phys. (2001)

Diabolical features of the low-lying electronic states of TiC2

(7a1)2 (4b2)2 (8a1)2 (9a1)2 (3b1)2 (10a1)1 (5b2)1

2s *2s 2p 2p 2p Ti(d) Ti(d)

Principal Configuration of 3B2 State of TiC2

N2-like C22- Two singly-occupied

d orbitals of Ti2+

Which two d orbitals are occupied?

{dz2, dxz, dyz, dx2-y2, dxy} → {d(a1), d(a1), d(b1), d(b2), d(a2)}

Two possible principal configurations for most triplet states of TiC2:3B2 – (a1,b2) or (b1,a2)3B1 – (a1,b1) or (b2,a2)3A2 – (a1,a2) or (b1,b2)3A1 – only (a1,a1)

These are not distinct states

TOTAL ENERGIES: -924.17202516 -924.00962060

CI vector -------------------------------------------- a1 b1 b2 a2 CI coefficients -------------------------------------------- 222+00 20 2+00 00 0.9296021 -0.0117561 2+2200 20 2+00 00 -0.0013861 -0.5578912 22+200 20 2+00 00 0.0021951 0.3644880 2+2-+0 20 2+00 00 -0.0256925 -0.3208740 2+2+-0 20 2+00 00 0.0127608 0.2761282 2-2++0 20 2+00 00 0.0199491 0.2023749 22+-+0 20 2+00 00 -0.0023326 0.1756278 22++-0 20 2+00 00 0.0082242 -0.1675529 2+2++0 20 2-00 00 -0.0070174 -0.1576292 22+200 20 20+0 00 -0.0131613 0.1491989 2+2200 20 20+0 00 -0.0026000 -0.1426208 220+00 20 2+20 00 -0.0995256 -0.0010133 222+00 00 2+00 20 -0.0960682 0.0051868 2022+0 20 2+00 00 -0.0092721 0.0925050 +22200 20 2+00 00 0.0000511 0.0892032 202+20 20 2+00 00 -0.0874160 -0.0001599…

Two Lowest 3B2 Electronic States from CAS(12,14) Calculation withInitial Orbitals from (a1,b2) SCF

The first excited 3B2 state does NOT have the (b1,a2)principal configuration…

It has mostly excited (a1,b2) configurations…

Similarly, the first excited 3B2 state from a CAS(12,14) calculation with initial orbitals from a (b1,a2) principal configuration does not have the (a1,b2) principal configuration…

Symmetry Princ. R(Ti-C) C-Ti-C Energy Rel. EnergyConfig. (Angstrom) Angle (Hartree) (kcal/mol)

3B2 b1,a2 2.0524 35.748 -925.36004397 10.833B2 a1,b2 1.9910 37.583 -925.37730359 0.00

3B1 b2,a2 1.9784 37.960 -925.37561238 1.063B1 a1,b1 2.0664 35.432 -925.35787060 12.19

3A2 a1,a2 2.0014 36.842 -925.37130947 3.763A2 b1,b2 2.0341 36.535 -925.34501916 20.26

B3LYP Triplet TiC2 Energies at Optimal GeometriesWachters(spd)+Bauschlicher(f),Ti; cc-pVTZ(spd),C

There is a group of low-lying triplet states of TiC2 that differ only by which two d orbitals of Ti are singly occupied…

For a given state symmetry with two possible occupations for the principal configuration, only one of the possible occupations describes the actual state…

Convergence of the SCF orbitals is complicated by local minima and depends on the initial orbital occupation, the molecular geometry and the initial guess…

The previous study didn’t quite get it right..

They apparently locked in on the (b1,a2) principal configuration of the 3B2 state,but they almost got the right answer for the wrong reason.

State Principal Configuration Dipolea Q(Ti)b 3B2 10a1

13b125b2

1 0.932 8.57 1.03 3B1 9a1

23b125b2

11a21 0.933 8.57 1.01

3A2 10a113b1

24b221a2

1 0.928 9.06 1.07 3A1 10a1

111a113b1

24b22 0.929 9.34 1.04

1A2 10a1( )3b124b2

21a2( ) 0.648 1.32 0.73

10a1 ( )3b124b2

11a2 ( ) -0.648

1A1 10a123b1

24b22 0.780 1.97 0.82

10a1011a1

23b124b2

2 -0.483 1B2 10a1

1( )3b125b2

1 ( ) -0.648 7.25 0.96

10a11( )3b1

25b21( ) 0.648

1B1 9a123b1

25b21( )1a2

1 ( ) -0.651 8.10 0.99

9a123b1

25b21( ) 1a2

1( ) 0.651

Principal Configurations, dipole moments (Debye), and charge distribution for different electronic states of TiC2 at CISD optimized geometries

a Calculated as an expectation value at the RMRCI level of theory.b Calcuated at CASSCF(12,14) level with uncontracted Wachters(spd)+Bauschlicher(f) basis set for Ti, and Dunning cc-pVTZ(spd) for C.

TiC2 Relative Energies (in cm-1) at Various Levels of TheoryRestrict (0,2) in MRCI

State Princ. CAS(12,14) MRCI(12,14) MRCI(12,14) MRCI(12,14)+Q C(0) in MRCI Rel. EnergyConfig. Energy Reference En. Energy Energy (kcal/mol)

Restrict(0,2) in MRCI

3B1 (a1,b1) -924.17549617 -924.15172284 -924.34653923 -924.36471087 0.95331917 3.83

3B1 (b2,a2) -924.17737130 -924.15326241 -924.35144025 -924.36975621 0.95371938 0.66

3B2 (a1,b2) -924.17202514 -924.14863663 -924.35212619 -924.37080767 0.95431676 0.00

3B2 (b1,a2) -924.17208164 -924.14841825 -924.33958525 -924.35729043 0.95453453 8.48

3A2 (a1,a2) -924.17981653 -924.14662502 -924.34421607 -924.36167434 0.95514093 5.73

Restrict(0,3) in MRCI

3B1 (b2,a2) -924.17737134 -924.16150205 -924.35477081 -924.37263670 0.95399433 0.53

3B2 (a1,b2) -924.17202516 -924.15714937 -924.35535422 -924.37348095 0.95481855 0.00

3A2 (a1,a2) -924.17981980 -924.16059089 -924.34904347 -924.36526139 0.95680549 5.16

No Restriction in MRCI

3B1 (b2,a2) -924.17737134 -924.17737134 -924.35738685 -924.37344703 0.95587273 0.39

3B2 (a1,b2) -924.17202516 -924.17202516 -924.35770654 -924.37406912 0.95651699 0.00

3A2 (a1,a2) -924.17981980 -924.17981980 -924.35195491 -924.36604828 0.95919042 5.03

Present MRCI Calculations of TiC2 Triplet States:Wachters(spd)+f, Ti; cc-pVTZ(spd), C

B3LYP Calculations of the TiC2 3B2 & 3B1 Potential Energy Surfaces (C2v)

The ground 3B2 state intersects the first excited 3B1 state along a line running approximately from (R, ) = (1.75, 35) to (2.05, 44)

B3LYP Calculations of the TiC2 3B2 & 3A2 Potential Energy Surfaces (C2v)

The ground 3B2 state intersects the excited 3A2 state along a line running approximately for (R, ) = (2.15, 28) to (1.75, 36)

State Symmetry Reps. Double Group Symmetry

3B2 C2v B2 (A2+B1+B2)B2 = B1 + A2 + A1

Cs A’ (A’+2A”)A’ = A’ + A” + A”

3B1 C2v B1 (A2+B1+B2)B1 = B2 + A1 + A2

Cs A” (A’+2A”)A” = A” + A’ + A’

3A2 C2v A2 (A2+B1+B2)A2 = A1 + B2 + B1

Cs A” (A’+2A”)A” = A” + A’ + A’

Conical Intersections in the Triplet Manifold of TiC2

The spin-orbit splitting of the ground 3F state of the Ti atom is 170.132 cm-1

It seems possible that the ground vibronic state of TiC2 is delocalizedover two or three Born-Oppenheimer potential energy surfaces…

The electronic ground state of the titanium metcar, Ti8C12

Ti(o)

Ti(o)

Ti(i)

Ti(i)

Ti(o)

DFT and most ab initio single-reference CI methods predict a triplet ground state…

The Last 4 Valence Electrons…

4e

3t1

6t2

4a1

4t17t2

5e5a1

Ti8C124+

4a1 7t2

4t1

Key issue in determining the molecular symmetry and ground state

Symmetry preserving quintet states

Fill the 4a1 orbital and consider the+2 ion

4 a1

4 e 3 t1

6 t2

4 t1

7 t2

8 t2

5 a1

Ti8C122+

LUMO

4t1 7t25a1

Bottom line: 2 electrons go into4a1 and 2 go into either the 4t1

or 7t2 orbital; Jahn-Teller distortionto a lower symmetry.

Symmetry preserving singlet state

Symmetry breaking triplet states

Symmetry breaking singlet states

The Last 2 Valence Electrons…

There are two states of interest in each symmetry, mainly different in the HOMO (either on Ti(i) dz2 or on Ti(o) dxy interacting with C-C * orbital)

HOMO1(4t1)

HOMO2(7t2)

Jahn-Teller Distortion: D2d & C3v symmetries

a

a Energies in kcal/mol relative to the lowest-energy configuration at the same level of theory.

Singlet state

1A1 State (in C2v subgroup of D2d) 3A2 State (in C2v subgroup of D2d) E = 7141.44085315 E = 7141.42092017

a1 b1 b2 a2 coef. a1 b1 b2 a2 coef. 2000 00 00 20 0.9249 2+00 00 00 +0 0.7613 2000 20 00 00 -0.2034 2000 +0 +0 00 -0.6229 2000 00 20 00 -0.2034 ++ 0 00 00 0+ 0.0548 2000 00 00 02 -0.0911 ++0 00 00 0+ -0.0548 +00 00 00 + 0.0545 0+0 +0 0+ 00 0.0502 + 00 00 00 + 0.0545 +00 00 00 + -0.0542 + 00 00 00 + -0.0542 000 0 0+ +0 -0.0523 +000 +0 0 0 -0.0523 000 0+ 0 +0 0.0523 +000 0 +0 0 0.0523

CAS(4,10) Comparison of Singlet & Triplet Energies

E(CAS) = 12.50 kcal/mol

49a1 0.205548a1 0.170947a1 0.1519(32b1,32b2) 0.138919a2 0.1133(31b1,31b2) 0.052318a2 -0.097646a1 -0.2441

a2 HOMO singlet a2b2 triplet

1A1 State (in C2v subgroup of D2d) 3A2 State (in C2v subgroup of D2d) EMRCI = 7241.453224

EMRCI+Q = 7241.453607 EMRCI = 7241.436362

EMRCI+Q = 7241.437069 a1 b1 b2 a2 coef. a1 b1 b2 a2 coef.

2000 00 00 20 0.8930 2+00 00 00 +0 0.7661 2000 20 00 00 0.2460 2000 +0 +0 00 0.5834 2000 00 20 00 0.2460 +0 0 +0 0+ 00 0.0670 + 00 00 00 + 0.1032 ++ 0 00 00 0+ 0.0594 +000 0 +0 0 0.0899 +000 + 00 +0 0.0587 2000 00 00 02 0.0828 +000 +0 0 0 0.0796 +000 0 +0 0 0.0514

Singlet & Triplet MRCI Energies with CAS(4,10) Reference Function

E(MRCI+Q) = 10.38 kcal/mol

Basis: Ti, Bauschlicher-ANO(spdf); C, 6-31G*(spd)

# internal configurations = 225# contracted configurations = 4,029,687

# internal configurations = 252# contracted configurations = 3,643,308

a2 HOMO singlet a2b2 triplet

CAS(16,14) 1A1 (b2 HOMO) CI vector ========================== a1 b1 b2 a2 coefficient ========================== 2220 2200 2200 20 0.7450422 2200 2200 2220 20 -0.5635310 2020 2220 2200 20 -0.0502075

TOTAL ENERGY -7241.53910301

a2 homob2 homoa2b2 tripletee triplet

a2 homob2 homoa2b2 tripletee triplet

b2 homoa2b2 triplet

becomes b2 homo

End

The work at Brookhaven National Laboratory was funded under contract DE-AC02-98CH10886 with the U.S. Department of Energy and supported by its Division of Chemical Sciences, Geosciences & Biosciences, Office of Basic Energy Sciences.