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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

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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

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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 + )

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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

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NiF 3d 9 Electronic Energy Levels 0 1000 2000 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

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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

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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

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(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)

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-3 -2 -1 0 1 2 0123456 2 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)

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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 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

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= 1/2 Rotationally analyzed Not rotationally analyzed NiF Electronic states J. Mol. Spectrosc. 214 (2002) 152-174 Krouti, Hirao, Dufour, Boulezhar, Pinchemel, Bernath 3d 9 manifold

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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 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) 154312

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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) 168-171 PdX (4d 9 ): Microwave spectroscopy T. Okabayashi’s group

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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)

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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)

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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.

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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

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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 4 + 25L z 2 – 6L 2 ]

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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) 7164-7178 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.

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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

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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

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