An approximate H eff formalism for treating electronic and rotational energy levels in the 3d 9 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 3d 9 manifold of electronic states Related molecules: PdX and PtX Configuration: 4d 9 5d 9 Topics considered: Position of all 3d 9 spin-orbit components Large  -type doubling in  = ½ states

Effective Hamiltonian = H non-rot + H rot H electronic = H Crystal-Field + H Spin-Orbit H CF = C 0 + C 2 Y 20 (  ) + C 4 Y 40 (  ) = (unfamiliar) H electronic-rotational = H CF + H SO + H rot H SO = A L·S (familiar)  = AL z S z + ½ A(L + S  + L  S + )

All 10 electronic basis set functions | ,  with L=2 and S=½ 2  5/2 |  2,  ½   =  5/2 2  3/2 |  2,  ½   =  3/2 2  3/2 |  1,  ½   =  3/2 2  1/2 |  1,  ½   =  1/2 2  1/2 |  0,  ½   =  1/2

NiF 3d 9 Electronic Energy Levels cm A   obs AL·SAL·S Ni + 2 D 2 D 5/2 2 D 3/2 1/2 5/2 1/2 2 D 2  2  2  A=0 mol. 3/2

Fit the observed electronic levels (= 5 spin-orbit components) to determine Two crystal-field splitting parameters = C 2 and C 4 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 + C 2  5C 4 )/4 rsin2  = +  A  (3/2) The two  = ½ wave functions become | ,  | ,   upper = +cos  |1,  ½  + sin  |0,+½   lower =  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  H rot = B(J  L  S) 2 = B[(J 2  J z 2 )+(L 2  L z 2 )+(S 2  S z 2 )]  2B[(J x S x +J y S y )+(J x L x +J y L y )] + 2B(L x S x +L y S y ) Taking red terms into account E rot (  =½) = BJ(J+1)  ½p(J+½)

 in radians p/2B forupperandlower  =1/2 states of NiCl with empirical correction factor  =0.89 in  =1|L + |  =0  2  calc obs p/2B (unitless)

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 = (signs different) (p/2B) experimental = (signs the same) Can be decided by experiment. An experimental test of this theory would be to analyze an appropriate pair of NiF transitions to determine the relative signs of p for the two  = 1/2 states in the 3d 9 manifold

 = 1/2 Rotationally analyzed Not rotationally analyzed NiF Electronic states J. Mol. Spectrosc. 214 (2002) Krouti, Hirao, Dufour, Boulezhar, Pinchemel, Bernath 3d 9 manifold 

NiCl Absolute parities (p/2B) theoretical = (signs – and +) (p/2B) experimental = (signs + and –) Can only be decided by theory. There are two theoretical results asking for this 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)

New experimental work to which this theory should be applicable NiI (3d 9 ): Electronic spectroscopy: V.L. Ayles, L.G. Muzangwa, S.A. Reid Chem. Phys. Lett. 497 (2010) PdX (4d 9 ): 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 d 9 manifold (probably quite easy with this model) Look at d 8 s manifold (maybe not doable with this model)

It is much less convenient to use the case (b) splitting expression E rot ( 2  ) = BN(N+1) + ½  N for J=N+1/2 E rot ( 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 E rot (  =½) = BJ(J+1)  ½p(J+½)

H CF = C 0 + C 2 Y 20 (  ) + C 4 Y 40 (  ) = unfamiliar Y l,m>0 ( ,  ) do not occur in electric field for a cylindrical symmetric charge. Y l >4,0 (  ) do not have non-zero matrix elements within L = 2 manifold. Y odd,0 (  ) do not have non-zero matrix elements within 3d manifold.

C 0 Y 00 is a constant energy shift C 2 Y 20 (  ) is interaction of charge of d-hole with electric quadrupole moment of the molecule C 4 Y 40 (  ) is interaction of charge of d-hole with electric hexadecapole moment of the molecule

Operator equivalents Greatly simplify calculations Good for  L = 0 matrix elements  Good within d 9 manifold C 2 Y 20 (  )  (1/6)C 2 [3L z 2 – L 2 ] C 4 Y 40 (  )  (1/48)C 4 [35L z 4 – 30L 2 L z 2 + 3L L z 2 – 6L 2 ]

3 T 0 ’s are used in the paper The electronic structure of NiH: The {Ni + 3d 9 2 D} supermultiplet. by J.A. Gray, M. Li, T. Nelis, R.W. Field, J. Chem. Phys. 95 (1991) We use 3 crystal-field parameters C 0,C 2,C 4 for 3 electronic states 2 , 2 , 2  ! Why not just use T 0 for each state??? I hope that variation of the C 0,C 2,C 4 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 C 2 =C 4 =0 (only spin-orbit interaction) Errors  4% of total 3d 9 manifold spread Predict 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 have 3 parameters (C 0, C 1, C 2 ) for 5 levels or 4 parameters (C 0, C 1, C 2,  ) for 5 levels