Electronic Supplementary Material (ESI for hysical Chemistry Chemical hysics. This journal is the Owner Societies 06 Supporting Information to Orientation order and rotation mobility of nitroxide biradicals determined by quantitative simulation of ER spectra. Matrix elements for rotation diffusion superoperator in the case of noncoincident orientation and rotation tensors. The goal of this section is to determine the matrix elements of stochastic iouville superoperator for the purpose of ER spectra simulation in the situation, when the principal axes of rotation diffusion are not coincident with the principal axes of molecule orientation. et us consider Smoluchowski equation for rotation diffusion in orientation potential [, ]: ( W,t W0,0 -Γ( W,t W0,0 t ÊU( W -R( W,t W0,0 -R ( W,t W 0,0 Ë kbt - R È ij i j + i( ju ( W,t W0,0 i, j xyz,, Here ( W,t W0,0 is the probability density to find the molecule at moment t in the orientation given by Euler angles W, on the condition that at zero time the molecule had the orientaition W 0, Γ is the stationary Markovian operator defining correlation functions for rotation, diffusion tensor is assumed to be symmetric, Rij Rji, is the operator of infinitesimal rotation, identical to dimensionless quantum-mechanical angular U( W momentum operator, U is dimensionless mean-field potential. kbt Equation ( defines stationary orientation distribution 0 ( W, such that Γ 0 ( W 0. Explicit expression for 0 ( W is given by Maxwell Boltzmann distribution: 0 ( W e- U For calculation of ER spectra, one should work with symmetrized superoperator -/ / Γ 0( W Γ 0( W ( (
which is self-adjoint. By applying symmetrization transformation ( to expression (, one obtains: ( Γ e Γe e R + e U / - U / U/ È / ij i j i j - U U (3 i, j xyz,, Since the angular momentum operator is the first order differential operator, the following identities can be used: ( ( + ( i f g f i g f g i -U / Ê / ( -U - ( iu i e e Ë By using expressions (, equation (3 transforms into È Γ Rij Í - iu ju + i ju + i ju - iu j + i j i, j xyz,, ( ( ( ( ( ( ( ( È Rij Í - i j + i j + i j i, j xyz,, (5 The second equality in (5 follows from the symmetry of diffusion tensor, Dij Dji. By ( U ( U ( U ( ( expanding the sum in (5, one obtains: È Γ RxxÍ - xu + x U + x È + Ryy Í - ( yu + ( y U + ( y È + Rzz Í - ( zu + ( z U + ( z È + Rxy Í - ( xu ( yu + (( x y + yx U + ( x y + yx È + Rxz Í - ( xu ( zu + (( x z + z x U + ( x z + z x È + Ryz - ( yu ( zu + (( y z z y ( y z z y Í + U + + ( ( ( (6 The expression (6 can be rewritten with the use of operators, z, ± x ± iy, for which the set of Wigner functions ( W D M with constant and M is closed. It is convenient to adopt the following notation for combinations of diffusion tensor components []:
( Rxz + iryz Rxx + Ryy Rxx - Ryy + irxy R r, e, h zz, k Rxx + Ryy Rxx + Ryy Rxx + Ryy (7 Then the equation (6 transforms into: e r er r hr Γ- ( + U - ( - U - ( + U ( -U - ( zu 8 8 e r er r ( h - r + ( + U + ( - U + ( U + ( z U e r er + + + - + r + ( h- rz (8 - ( + U ( zu - ( -U ( zu + È( + z + z + È( - z + z - U+ U + ( + z + z + + ( - z + z - The items in (8 can be divided into two groups. The first group contains the terms, which are independent of the mean-field potential: e r er Γiso + + - + r + ( h - rz (9 + ( + z + z + + ( - z + z - The remaining terms, i. e. those depending on mean-field potential, comprise the second group: Γ U e r er r hr - ( + U - ( - U - ( + U ( -U - ( zu 8 8 e r er r ( h - r + ( + U + ( - U + ( U + ( z U (0 - ( + U ( zu - ( -U ( zu + È( + z + z + È( - z + z - U+ U The operator Γ will be calculated in the basis of normalized Wigner functions:
M M + D M W 8p ( + D M W 8p ( Then the corresponding matrix elements are given by ( + ( + Γ Ú M M ( D ( ΓD ( ( M M dw W W 8p The following properties of the angular momentum operators are used for calculation of matrix elements: ( W È ( + - ( ± È ( + -( ± ( ± ( W ± DM D M± (3 ( W ( + - ( ± D ( W È ± z DM M± ( W ( ± ( + - ( ± D ( W È z ± DM M± Also, the following notation is used for eigenvalues of ladder operators: ( ( M± (, + - ± (5 By applying Eqs. (, (3 and ( to expression (9, one obtains the matrix elements of the superoperator Γ iso : ( M Γ iso M { ( ( d d MM d È r + + h - r È + d ( + Í + M+ (, Í È + d ( -Í - M-(, È e r + d + Í M+ (, M+ (, + Í Èer + d -Í M-(, M-(, - (6 et us now turn to calculating matrix elements for potential-dependent terms Γ U. For the mean-field potential, the decomposition in Wigner functions is used
U - aq D0 q( W (7 0 q- Only terms with even are non-zero in (7, due to the fact that the mean-field potential is centrosymmetric. The superoperator Γ U can also be decomposed into the series in Wigner functions: Γ X D ( W (8 U q 0q 0 q- For the components X q the same selection rules hold as for the potential decomposition terms a q. The matrix elements, which correspond to equation (8, are given by ( + ( + X M Γ U M 0-8p ( d È ( ( Ú W ÍD W ΓD W D0 W M M ( + ( + X - 0 M+ Ê Ê (- ËM 0 -M Ë - - The last equality in (9 was obtained with the use of Clebsch-Gordan decomposition: D ( D ( W W MN MN + M+ N Ê Ê - + M MN W -M M N -N N - MN, - Ë Ë ( ( D ( (9 (0 For calculating the terms Wigner functions, as follows: X, individual terms in (0 should be expanded in series in
± U - aq M± q 0q± W 0q- U ( D ( - aq + 0q W q (, D ( ± U q q ± ± ( 0q± 0q± q q q ( - ( + D0 q ( W a a M (, (, q q ± q M± q q, q q Ê Ê 0 0 0 q± -q q ± Ë Ë ( a a M (, q M (, q D ( W D ( W + - q q + - 0q+ 0q- q q q - + 0 q W q q q, q q + - Ê Ê Ë 0 0 0 Ëq+ -q q - ( U ( U a a M (, q M (, q D ( W D ( W zu - aq qd0q W 0q- ( ( D ( a a M (, q M (, q ( z U - aq q D 0 q W 0q- ( zu (3 q q 0q 0q q q q ( - ( + D0 q ( W a a q q q q q, q q Ê Ê 0 0 0 q -q q Ë Ë ( a a q q D ( W D ( W (
( U ( U a a M (, q q D ( W D ( W + z q q + 0q+ 0q q q q ( - ( + D0 q ( W a a M (, q q + q q q, q q Ê Ê 0 0 0 q+ -q q Ë Ë ( U ( U a a M (, q q D ( W D ( W - z q q - 0q- 0q q q q (- ( + D0 q ( W a a M (, q q - q q q, q q Ê Ê 0 0 0 q- -q q Ë Ë (, (, D ( ± U - aq M± q M± q+ 0q± W q (, (, D ( - aqm M± qm M± qm 0q W q ( + U - a M (, q( q± D ( W ± z z ± q ± 0q± q (, ( D ( - aqm M± qm qm 0q W q ( (5 (6 Then, by combining expressions (0, ( (6, the overall value of as: X can be written
X r ( ( h -aq È + + - q e r - aq- M+ q- M+ q- er (, (, (, (, - aq+ M- q+ M- q+ - aq- M+ ( q, -( q- (, ( - aq+ M- q+ q+ q Ê -(- ( + a a q q 0 0 0 q q Ë È e r Ê Í M+ (, q M+ (, q Í 8 Ëq+ -q q + er Ê + M-(, q M-(, q 8 Ëq- -q q - r Ê + M+ (, q M-(, q Ëq+ -q q - hr Ê + q q Ëq -q q Ê + M+ (, q q Ëq+ -q q Ê + M-(, q q Ëq- -q q (7 Now one can return to designations D ij for the rotation diffusion tensor components. For the matrix elements of superoperator Γ iso, one obtains:
M Γ iso M ÔÏ ÈRxx + Ryy d d ( MM Ìd È Í + - + Rzz Ô Ó ÈRxz -iryz + d ( + Í + M+ (, ÈRxz + iryz (8 + d ( -Í - M-(, ÈRxx -Ryy -irxy + d + Í M + (, M + (, + + d ÈR - R + ir Ô M (, M (, - Ô xx yy xy - Í - - Following the ideas of Refs. [, 3, ], for rendering of stochastic iouville superoperator in the complex symmetric form, the following modified basis should be used: + ( ( j Mj M + j - M - ( + d0 where (9 j sgn(, if 0 j -, if 0 ( (30 It can be shown that in the modified basis, the matrix elements of superoperator Γ iso are given by the following equations:
if, j j M j Γ iso M j Re M Γiso M -d (, j - Re M - Γiso M (3 if, j j ( iso d, - Im - iso M j Γ M j M Γ M (3 if, j j ( d ( d / Γ iso È + 0 + 0 Re Γ iso M j M j M M (33 if, j j ( d ( d / M j Γiso M j - j È + 0 + 0 Im M Γiso M (3 where M Γ iso M ÔÏ ÈRxx + Ryy d d ( MM Ìd È Í + - + Rzz Ô Ó ÈRxz -iryz + d ( + Í + M+ (, ÈRxz + iryz + d ( -Í - M-(, ÈRxx -Ryy -irxy + d + Í M + (, M + (, + + d ÈR - R + ir Ô M (, M (, - Ô xx yy xy - Í - - One can see that in the case of diagonal diffusion tensor, equations (3 (3 are identical to the equation (B33 in Ref. [3]:
M j Γiso M j Ï Ô ÈRxx + Ryy d d ( MM d j j Ìd È Í + - + Rzz Ô Ó Rxx -Ryy / + Èd - N ( +, d + N ( -, È( d 0( d0 + + + where N (, M (, M (, ±. ± ± ± (35 By returning to designations D ij for the rotation diffusion tensor components, the following expression for the spherical components of superoperator Γ U are obtained: X q aq ÊRxx + Ryy - È ( + - q + Rzzq Ë Rxx -Ryy -irxy aq- - M+ q- M+ q- Rxx - Ryy + irxy aq+ - M- q+ M- q+ Rxz -iryz aq- - M+ (, q-( q- Rxz + iryz aq+ - M-(, q+ ( q+ q + Ê -(- a a q q Ë 0 0 0 q q (, (, (, (, ÈRxx -Ryy -irxy Ê Í M+ (, q M+ (, q q q q Ë + - + Rxx - Ryy + irxy Ê + M-(, q M-(, q q q q Ë - - - Rxx + Ryy Ê + M+ (, q M-(, q q q q Ë + - - Ê + Rz q q q q q Ë - Ê + ( Rxz -iryz M+ (, q q q q q Ë + - Ê + ( Rxz + iryz M-(, q q q q q Ë - - (36
One can see that in the case of diagonal rotation diffusion tensor, the equation (36 reduces to Eq, (B6 in Ref. [3]: X ( a - N+ (, q a + N-(, q Ê Rxx -Ryy q - q - + q + + Ë Ê Rxx + Ryy + aq È ( + - q + Rzzq - Ë + Ê - a a q q 0 0 0 q q Ë ÈRxx -Ryy Ê Ê Í M+ (, q M+ (, q + q+ -q q + Í Ë Ë Ê + M-(, q M-(, q q- q q + Ë - - Rxx + Ryy Ê + M+ (, q M-(, q q q q + Ë + - - Ê + Rzz q q q q q Ë - For calculation of matrix elements of superoperator Γ U in the modified basis (9, the expression (9 should be modified. For this, the following symmetry relation for should be noted: - q - ( q ( X X (37 q which follows from the symmetry property of the mean-field potential: ( q ( a a- q - q (38 This entails the symmetry property for the matrix element: ( + + + M - Γ U M - - M Γ U M Thus, any matrix element in symmetrized basis (9, can be calculated from the matrix elements in the original basis (: X
/ - M j Γ M j Ê j j È ( + d 0( + d0 Ë ( M Γ M + j j M Γ M + j - M M - + j j M M - + ( ( Γ Γ if j j ( d ( d / - M j Γ M j È + 0 + 0 if j j + ( Re M Γ M j ( Re M Γ M + - - / - Γ È( + d ( + d ( Im + M Γ M j ( ( Im M Γ M M j M j j 0 0 + - - (39 (0 In the explicit form, if j j ( d ( d ΓU M j È + 0 + 0 M j -/ M+ Ê ( + ( + ( - M 0 0 M Ë - È Ê ÍRe X - Ë - - + Ê + j ( - Re X + - + - Ë ( if j j
( d ( d M j M j j ΓU È + 0 + 0 M+ Ê ( + ( + ( - M 0 0 M Ë - È Ê ÍIm X - Ë - - + Ê + j ( - Im X + - + - Ë -/ ( The equation ( is identical to Eq. (B7 from Ref. [3]. That means, that for diagonal rotation diffusion tensor, and orthorhombic potential, expressions ( ( reduce to those known in literature.
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