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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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DMITRY G. MELNIK AND TERRY A. MILLER The Ohio State University, Dept. of Chemistry, Laser Spectroscopy Facility, 120 W. 18th Avenue, Columbus, Ohio 43210.

DMITRY G. MELNIK AND TERRY A. MILLER The Ohio State University, Dept. of Chemistry, Laser Spectroscopy Facility, 120 W. 18th Avenue, Columbus, Ohio 43210.

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