An approximate Heff formalism for treating electronic and rotational energy levels in the 3d9 manifold of nickel halide molecules

Jon T. Hougen NIST

Molecules considered:

NiX = NiF, NiCl, NiBr, and NiI (No NiH)

Ni+X has one “d hole” has 3d9 manifold of electronic states

Related molecules: PdX and PtXConfiguration: 4d9 5d9

Topics considered:

Position of all 3d9 spin-orbit componentsLarge -type doubling in = ½ states

Effective Hamiltonian = Hnon-rot + Hrot

Helectronic = HCrystal-Field + HSpin-Orbit

HCF = C0 + C2Y20() + C4Y40() = (unfamiliar)

Helectronic-rotational = HCF + HSO + Hrot

HSO = A L·S (familiar) = ALzSz + ½ A(L+S + LS+)

All 10 electronic basis set functions |, with L=2 and S=½

25/2 |2, ½ = 5/223/2 |2, ½ = 3/223/2 |1, ½ = 3/221/2 |1, ½ = 1/221/2 |0, ½ = 1/2

NiF 3d9 Electronic Energy Levels

0

1000

2000

cm-1

A obs

AL·S Ni+

2D

2D5/2

2D3/2

3/2

1/2

5/2

1/2

2D

2

2

2

A=0mol. 3/2

NiCl 3d9 Electronic Energy Levels

0

1000

2000c

m-1

2D3/2

2D5/2

2D 2D

2

2

2

1/2

1/2

3/2

3/2

5/2

A=0mol.

A

obs

AL·Satom

Fit the observed electronic levels (= 5 spin-orbit components) to determine

Two crystal-field splitting parameters = C2 and C4 and

One spin-orbit splitting parameter = A and One “orbital impurity factor” 0.9

Turn these four parameters into one=0 mixing coefficient parameter

Define: rcos2 = (A + C2 5C4)/4 rsin2 = +A(3/2)

The two = ½ wave functions become |, |, upper = +cos |1,½ + sin |0,+½lower = sin |1,½ + cos |0,+½

L=2,=1|L+|L=2,=0 = [L(L+1)]1/2

(p/2B)upper = ½ + ½cos2 6 sin2(p/2B)lower = ½ ½cos2 + 6 sin2

Hrot = B(JLS)2

= B[(J2Jz2)+(L2Lz

2)+(S2Sz2)]

2B[(JxSx+JySy)+(JxLx+JyLy)] + 2B(LxSx+LySy)

Taking red terms into account

Erot(=½) = BJ(J+1) ½p(J+½)

-3

-2

-1

0

1

2

0 1 2 3 4 5 6

p/2B

(uni

tles

s)

2 in radians

p/2B for upper and lower =1/2 states of NiF with empirical correction factor =0.875 in =1|L+|=0

2calc

obs

obs

-3

-2

-1

0

1

2

0 1 2 3 4 5 6

2 in radians

p/2B for upper and lower =1/2 states of NiCl with empirical correction factor =0.89 in =1|L+|=0

2calc

obs

obs

p/2B

(u

nitle

ss)

What questions have been raised by this theory?

Main questions concern parity assignments (+ or -) of the rotational levels, which affect sign of p.

NiF Relative parities (p/2B)theoretical = -2.51 +1.51 (signs different)(p/2B)experimental = -2.23 -1.19 (signs the same)Can be decided by experiment.

An experimental test of this theory would beto analyze an appropriate pair of NiF transitionsto determine the relative signs of p for the two = 1/2 states in the 3d9 manifold

= 1/2

= 1/2

Rotationally analyzed

Not rotationally analyzed

NiF Electronic states

J. Mol. Spectrosc. 214(2002) 152-174

Krouti, Hirao, Dufour,Boulezhar, Pinchemel, Bernath

3d9 manifold

NiCl Absolute parities (p/2B)theoretical = -2.46 +1.46 (signs – and +)(p/2B)experimental =+2.32 -1.32 (signs + and –)Can only be decided by theory.

There are two theoretical results asking forthis absolute parity sign change

1. The present work wants signs of p changed.

2.Ab initio work wants a 2+ state at 12,300 cm-1 to be reassigned as 2 W.-L. Zou & W.-J. Liu, J. Chem. Phys.124 (2002) 154312

New experimental work to which this theory should be applicable

NiI (3d9): Electronic spectroscopy: V.L. Ayles, L.G. Muzangwa, S.A. Reid Chem. Phys. Lett. 497 (2010) 168-171

PdX (4d9): Microwave spectroscopy T. Okabayashi’s group

Possible new theoretical work

Formulas for splitting (J+1/2)3 in the two = 3/2 states of the d9 manifold (probably quite easy with this model)

Look at d8s manifold(maybe not doable with this model)

It is much less convenient to use the case (b) splitting expression

Erot(2) = BN(N+1) + ½N for J=N+1/2Erot(2) = BN(N+1) - ½(N+1) for J=N-1/2

Note that to treat all = ½ states on an equal footing, it is most convenient to use the case (a) splitting expression

Erot(=½) = BJ(J+1) ½p(J+½)

HCF = C0 + C2Y20() + C4Y40() = unfamiliar

Yl,m>0(,) do not occur in electric field for a cylindrical symmetric charge. Yl >4,0() do not have non-zero matrix elements within L = 2 manifold.Yodd,0() do not have non-zero matrix elements within 3d manifold.

C0Y00 is a constant energy shift

C2Y20() is interaction of charge of d-hole with electric quadrupole moment of the molecule

C4Y40() is interaction of charge of d-hole with electric hexadecapole moment of the molecule

Operator equivalentsGreatly simplify calculationsGood for L = 0 matrix elements Good within d9 manifold

C2Y20() (1/6)C2[3Lz2 – L2]

C4Y40() (1/48)C4[35Lz

4 – 30L2Lz2 + 3L4

+ 25Lz2 – 6L2]

3 T0’s are used in the paper

The electronic structure of NiH: The {Ni+ 3d9 2D} supermultiplet.by J.A. Gray, M. Li, T. Nelis, R.W. Field, J. Chem. Phys. 95 (1991) 7164-7178

We use 3 crystal-field parameters C0,C2,C4 for 3 electronic states 2,2,2 ! Why not just use T0 for each state???

I hope that variation of the C0,C2,C4 crystal-field parameters with halogen (F, Cl, Br, I) and with metal (Ni, Pd, Pt) will be more chemically meaningful than changes in energy positions.

Strengths of present electronic model:

We can visualize “2” limiting cases: A = 0 (no spin orbit interaction) or C2=C4=0 (only spin-orbit interaction)

Errors 4% of total 3d9 manifold spreadPredict 2 missing levels from 3 obs ??

Errors 0.4% with correction factor 0.9

L=2,=+2|L+|L=2, =+1 = [(L-1)(L+2)]1/2

L=2,=+1|L+|L=2, =0 = [L(L+1)]1/2

Weakness of present electronic model =too many adjustable parameters

Even with A = 603 cm-1 = fixed, we have3 parameters (C0, C1, C2) for 5 levels or4 parameters (C0, C1, C2, ) for 5 levels