1. Tom Ziegler Department of Chemistry University of Calgary,Alberta, Canada T2N 1N4 Magnetically Perturbed Time Dependent Density Functional Theory. Applications and Implementations Tuesday November 11 11:30 am - 12:10 pm

2. ADF Solves Kohn-Sham equations Properties –NMR, EFG, EPR, Raman, IR, UV/Vis, NLO, CD, … –Potential energy surfaces (transition states, geometry optimization) Environment effects –QM/MM, COSMO Relativistic effects –Scalar relativistic effects, spin-orbit coupling –Transition and heavy metal compounds Uses Slater functions

3. Inorganic Spectroscopy

4. Basic Time Dependent Density Functionl Theory Basic Equation : Definition of A and B Matrices : M.E.Casida Gross,E.K.; Kohn W. Where : T. Ziegler,M.Seth,M.Krykunov,J.Autschbach A Revised Electronic Hessian for Approximate Time-Dependent Density Functional Theory SUBMITTED, J.C.P. T. Ziegler,M.Seth,M.Krykunov,J.Autschbach A Revised Electronic Hessian for Approximate Time-Dependent Density Functional Theory SUBMITTED, J.C.P.

5. Basic Time Dependent Density Functionl Theory Basic Equation : Corredted Definition of A and B Matrices : M.E.Casida Gross,E.K.; Kohn W. Where :

6. Basic Time Dependent Density Functionl Theory Basic Equation : Corredted Definition of A and B Matrices : M.E.Casida Gross,E.K.; Kohn W. Where : Spin-flip transitions using non-collinear functionals Liu (2004),Ziegler+Wang (2005),Vahtras (2007)

7. Basic Time Dependent Density Functionl Theory Transition Energy : Basic Equation : M.E.Casida Gross,E.K.; Kohn W. Electric Transition Dipole Moment : Magnetic Transition Dipole Moment :

8. A C B A B C Absorption Spectra and TD-DFT Transition Energy :

9. Inorganic Spectroscopy H

11. Why MCD and MOR ? Magnetic Circular Dichroism (MCD) Spectroscopy More information about each excited state In absorption spectroscopy only positive (often overlapping) bands In absorption spectroscopy only positive (often overlapping) bands

12. Why MCD ? Magnetic Circular Dichroism (MCD) Spectroscopy In MCD bands of different shapes More information about each excited state In MCD bands of different shapes More information about each excited state

13. Magnetic Circular Dichroism (MCD) Spectroscopy Origin of MCD ? Electric dipole operator: Absorbance in dipole approximation.

14. Magnetic Circular Dichroism (MCD) Spectroscopy Origin of MCD ? Absorbance in dipole approximation.

15. Magnetic Circular Dichroism (MCD) Spectroscopy Electric dipole operator For circular polarized Light: Electric dipole operator For circular polarized Light: Difference in absorbance of left and right circular polarized light Circular Polarized Light Origin of MCD ?

16. Magnetic Circular Dichroism (MCD) Spectroscopy The difference in absorption of left and right circularly polarized light in the presence of a magnetic field as a function of photon energy The difference in absorption of left and right circularly polarized light in the presence of a magnetic field as a function of photon energy Origin of MCD ?

17. Magnetic Circular Dichroism (MCD) Spectroscopy Origin of MCD ?

18. Magnetic Circular Dichroism (MCD) Spectroscopy The MCD disprsion P.J.Stephens. Ph.D. Thesis 1964 A B C( T ) AA

19. Magnetic Circular Dichroism (MCD) Spectroscopy The MCD disprsion P.J.Stephens. Ph.D. Thesis 1964 Absorption band Degenerate ground- or (and) excited state

20. Magnetic Circular Dichroism (MCD) Spectroscopy The MCD disprsion P.J.Stephens. Ph.D. Thesis 1964 Absorption band All cases

21. Magnetic Circular Dichroism (MCD) Spectroscopy The MCD disprsion P.J.Stephens. Ph.D. Thesis 1964 Absorption band Space and(or) spin-degenerate ground state Negative C-term Positive C-term

22. Origin of B -Term The B term M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

23. Expression for the B -Term The B term Or by using the identity We thus have M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

24. The Calculation of the B -term The B term : practical calculations We have: TD-DFT calculations Where: Early work: J.Michl, J.Am.Chem.Soc. 100,6801 (1978) Early work: J.Michl, J.Am.Chem.Soc. 100,6801 (1978)

25. The B term : practical calculations We have: TD-DFT calculations Solve: Where: The Calculation of the B -term

26. The B term : practical calculations The B term : practical calculations By differentiation of The Calculation of the B -term M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

27. Here: Affords The Calculation of the B -term M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

28. The B term : practical calculations The B term : practical calculations The Calculation of the B -term Seth+Ziegler JCP,2008,in press M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

29. The B term : Direct method We must solve The Calculation of the B -term by Direct Method Seth+Ziegler JCP,2008

30. The B term : Direct Method We must solve Pros Cons The Calculation of the B -term by Direct Method Seth+Ziegler JCP,2008,in press M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

31. The B term : Sum Over State Or Writing Z (1) in terms of the complete set F (0) affords The Calculation of the B -term by Sum-over-State Method Seth+Ziegler JCP,2008,134108 M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem. Phys. J. Chem. Phys. 128, 144105 (2008)

32. The B term : Sum Over State Pros Cons The Calculation of the B -term by Sum-over-State Method Seth+Ziegler JCP,2008,134108

33. Other B -term implementations 7S.Coriani, P.Jørgensen, T.Helgaker J.Chem.Phys. 113,3561,2000 HF+CI CCSD(T) E.Dalgaard Phys.Rev. A 42 42 1982 J.Olsen; P. Jørgensen J.Chem.Phys. 82 3235 (1985) W.A.Parkinson; J.Oddershede J.Chem.Phys. 94,7251 (1991) W.A.Parkinson; J.Oddershede) Int.J.Quantum Chem. 64,599 (1997) T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007)) DFT H.Solheim; L.Frediani; K.Rudd; S.Coriani Theor.Chem.Acc 119,231,2007 DFT-SOS M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107 M.Seth,T.Ziegler,J.Autschbach J.Chem.Theory.Comp.3,434,2007 J.Michl J.Am.Chem.Soc. 100, 6801, 1978 M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 144105 (2008)

34. Convergence of SOS- method for Ethylene Convergence of SOS- method for Ethylene Comparison of Sum-over-State and Direct Method for B -terms Seth+Ziegler JCP,2008

35. Comparison of Direct Method for B -terms with Experiment Exp: J.W.Waluk, J.Michl Inorg.Chem. 21,556,1982)Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986) S4N3+S4N3+ S4N2S4N2

36. Comparison of Direct Method for B -terms with Experiment Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986) Seth+Ziegler JCP,2008

37. Comparison of Direct Method for B -terms with Experiment and other Methods Exp: H.-P.Klein, R.T. Oakley, J.Michl Inorg.Chem. 25,3194 (1986) T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007))

38. TD-DFT calculations of B-term. Furan Thiophene Selenophen Tellurophen W. Hieringer, S. J. A. van Gisbergen, and E. J. Baerends J. Phys. Chem. A 2002, 106, 10380 Seth+Ziegler JCP,2008,134108

39. TD-DFT calculations of B-term. Sum-over-state formulation Norden, B.; Hansson, R.; Pedersen, P. B.; Thulstrup, E. W. Chem.Phys. 1978, 33, 355.

40. TD-DFT calculations of B-term. Sum-over-state formulation Norden, B.; Hansson, R.; Pedersen, P. B.; Thulstrup, E. W. Chem.Phys. 1978, 33, 355.

41. TD-DFT calculations of B-term. Sum-over-state formulation Furan Seth+Ziegler JCP,2008,134108

42. TD-DFT calculations of B-term. Sum-over-state formulation Thiophene Seth+Ziegler JCP,2008,134108

43. TD-DFT calculations of B-term. Sum-over-state formulation Selonophene Seth+Ziegler JCP,2008,134108

44. TD-DFT calculations of B-term. Sum-over-state formulation Tellurophen Seth+Ziegler JCP,2008,134108

45. A -term of MCD Origin of A -term M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943 M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)

46. The A -term of Magnetic Circular Dichroism (MCD) Spectroscopy The A term Thus M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943 M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)

47. The A -term of Magnetic Circular Dichroism (MCD) Spectroscopy The A term We have Thus M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008) Here

48. The A term The A -term of Magnetic Circular Dichroism (MCD) Spectroscopy

49. Other A -term implementations 7S.Coriani, P.Jørgensen, T.Helgaker J.Chem.Phys. 113,3561,2000 HF+CI CCSD(T) Y.Honda, M.Hada, M.Ehara, H.Nakatsuiji,J.Downing,J.Michl, Chem.Phys.Lett 355,219,, 2002 T.Kjœrgaard, B.Jansik, P.Jørgensen,S.Coriani, J.Michl, J.Phys.Chem. A 111,11278 (2007)) DFT H.Solheim; ; K.Rudd; S.Coriani,P.Norman J.Chem.Phys. 128,094193,2008 J.Michl J.Am.Chem.Soc. 100, 6801, 1978 Y.Honda, M.Hada, M.Ehara, H.Nakatsuiji,J.Michl, J.Chem.Phys. 123,164113 (2005) M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107 M.Seth,T.Ziegler, E.J.Baerends J.Chem.Phys. 2004,120,10943 M.Seth,T.Ziegler, J.Chem.Phys. 2007,127,134108 M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)

50. Applications: A/D Se 4 2+ Te 4 2+ Fe(CN) 6 4- Ni(CN) 4 2- C 6 Cl 6 C 6 H 3 Br 3 OhOh D 4h D 6h D 3h Exp: 0.72Calc: 0.63 Exp: 0.60Calc: 0.55 Exp: 0.40Calc: 0.48 Exp:-0.66Calc:-0.72Exp:-0.50Calc:-0.80 M.Seth,T.Ziegler,J.Chem.Phys. 2004,120,10943

51. Different MCD-terms Absorption band M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)

52. MCD-terms for Oxyanions M.Seth,T.Ziegler, M.Krykunov, J.Autschbach J.Chem.Phys. J. Chem. Phys. 128, 234102 (2008)

53. Exp.Theor MCD-terms for Thioanions M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. Submitted

54. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125. MCD spectra of Porphyrins containing Mg,Ni and Zn

55. Orbital level diagram for ZnP 2a 2u 1a 1u 1b 2u E.J. Baerends, G. Ricciardi, A. Rosa, S.J.A van Gisbergen J.Phys.Chem. A2001,105,3311 E.J. Baerends, G. Ricciardi, A. Rosa, S.J.A van Gisbergen Coord.Chem.Rev. 2002,230,5 2e g2 2e g1

56. Experimental Spectrum for ZnP E.J. Baerends, G. Ricciardi, A. Rosa, S.J.A van Gisbergen Coord.Chem.Rev. 2002,230,5 E.J. Baerends, G. Ricciardi, A. Rosa, S.J.A van Gisbergen Coord.Chem.Rev. 2002,230,5

57. Experimental Spectrum for ZnP L.Edwards,D.H.Dolphin,M.Goutermn J.Mol.Spectrosc 35(1970)90 E.J. Baerends, G. Ricciardi, A. Rosa, S.J.A van Gisbergen Coord.Chem.Rev. 2002,230,5

58. Simulated Spectrum for ZnP with A-term only Simulated Spectrum for ZnP with A-term only ZnP Exp A-only 1E u 2E u +3E u Q Q S Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125.

59. Influence of ring distortion on MCD spectrum of ZnP

60. Dist C 2V D 4h

61. Simulated Spectrum for ZnP with B-term only Exp. 1Eu 2Eu 3Eu B-terms

62. Simulated Spectrum for ZnP with A+B-term only Simulated Spectrum for ZnP with A+B-term only Exp. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125.

63. Simulated Spectrum for MgP and NiP with A+B-term 1Eu 2Eu 3Eu

64. Substituted Porphyrins MTPP MOEP tetraphenylporphyrinoctaethylporphyrin Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125.

65. Excited States for Substituted Porphyrins Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125.

66. Excited States for Substituted Porphyrins Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2007,46, 9111-9125.

67. Tetraazaporphyrins and Phthalocyanines

69. Alejandro Gonzales, Mike Seth, Tom Ziegler Inorg.Chem. Inorg. Chem. 2008,46, 9111-9125.

70. Magnetic Circular Dichroism (MCD) Spectroscopy The C term If

71. Electron configuration t 1u 6 t 2u 6 t 1u 6 t 2g 5 Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412

72. Limitations of Traditional TD-DFT What do we do with a degenerate ground state that can not be represented by single Slater determinant ? What do we do with a degenerate ground state that can not be represented by single Slater determinant ? Degenerate Ground State What are the fundamental equations ? What are the fundamental equations ? How do we calculate excitation energies How do we calculate excitation energies

73. TRICKS of the Trade: Calculating the Excitation Energies of Molecules with Degenerate Ground States using TD-DFT Degenerate ground states are generally treated within DFT by fractional occupations of the degenerate orbital. This gives a ground state of indeterminent symmetry. Degenerate ground states are generally treated within DFT by fractional occupations of the degenerate orbital. This gives a ground state of indeterminent symmetry. Challenges A degenerate ground state can be made non-degenerate by breaking utilizing a lower symmetry point group. The amount of symmetry breaking in this case can be large and symmetry assignments complicated A degenerate ground state can be made non-degenerate by breaking utilizing a lower symmetry point group. The amount of symmetry breaking in this case can be large and symmetry assignments complicated

74. Transformed Reference with an Intermediate Configuration Kohn Sham (TRICKS) TDDFT Solution: TRICKS of the Trade: Calculating the Excitation Energies of Molecules with Degenerate Ground States using TD-DFT Idea: Avoid problems with a degenerate ground state by taking an excited state that is nondegenerate as the (Transformed) Reference Intermediate Configuration.

75. Example 1: d 1 transition metal complexes of O h symmetry, d-d transition Example 1: d 1 transition metal complexes of O h symmetry, d-d transition Application of the TRIC method

76. Results 1: d 1 transition metal complexes of O h symmetry, d-d transition. Results 1: d 1 transition metal complexes of O h symmetry, d-d transition. Application of the TRIC method

77. Example 2: d 1 transition metal complexes of T d symmetry, d-d transition Example 2: d 1 transition metal complexes of T d symmetry, d-d transition Application of the TRIC method

78. Result 2: d 1 transition metal complexes of T d symmetry, d-d transition Result 2: d 1 transition metal complexes of T d symmetry, d-d transition Application of the TRIC method

79. Example 3: d 1 transition metal complexes of T d symmetry, charge transfer Example 3: d 1 transition metal complexes of T d symmetry, charge transfer Application of the TRIC method

80. Result 3: d 1 transition metal complexes of T d symmetry, charge transfer Result 3: d 1 transition metal complexes of T d symmetry, charge transfer Application of the TRIC method

81. Application: Fe(CN) 6 3- Electron configuration t 1u 6 t 2u 6 t 1u 6 t 2g 5 Excitations are ligand-metal charge transfer. C term of a transition to a T 1u state is positive and to a T 2u state is negative. TransitionExp.Calc. 11.21/0.610.86 2-0.68-0.86 30.560.86 Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412

82. More Applications RuCl 6 3- [Fe(CN) 5 SCN] 3- MnPc Exp.Calc. 0.580.84 -0.60-0.84 ExpCalc 7.57.3 6.97.3 -6.9-7.3 6.37.3 -3.1-7.3 2.27.3 Exp.Calc. 0.030.90 0.230.90 Seth,Ziegler,Autschbach,Ziegler JCP, 2005,09412

83. |A> |J> |K>  JA  KJ |A> |J> |K> |A> |J> |K>  KJ Spin-degenerate Ground State MCD via Spin-orbit Coupling M.L.Kirk Curr.Op.Chem.Bio 2003,220

84. Application to Plastocyanin

85. 85 § M.E. I. Solomon, R.K. Szilagyi, S. D. George and L. Basumallick, Chem. Rev, 104, 419, 2004. Application to Plastocyanin  KJ |A> |J> |K>

86. Application to Sulfite Oxidase

87. 87 § M.E. Helton, A. Pacheco, J. McMaster, J.H. Enemark and M. Kirk, J. Inorg. Biochem., 80, 227, 2000. Application to Sulfite Oxidase L 1 : -SCH 3. L 2 : -OH. L 3 : -S(CH 2 ) 2 S-. |A> |J> |K>  KJ

89. TD-DFT/MCD Dr. Mike Seth Dr.Jochen Autschbach Alejandro Gonzalez Peralta Dr. Mykhaylo Krykunov Fan Wang Hristina Zhekova

91. MOR and MCD` TD-DFT formulation without damping To obtain the solution From which we obtain density change in frequency domain With:

92. MOR and MCD The expression Allows us to calculate the MOR parameter V(  ) from the MCD parameters B J after summing over all states Allows us to calculate the MOR parameter V(  ) from the MCD parameters B J after summing over all states a M. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,

93. MOR and MCD The expression V res (  We need a TD-DFT formulation in which damping included

94. MOR and MCD` TD-DFT formulation with damping To obtain finite lifetime solutions To obtain finite lifetime solutions From which we obtain density change in frequency domain With: L.Jensen; J.Autchbach; G.C.Schatz J.Chem.Phys.2005,122,224115

95. MOR and MCD` TD-DFT formulation with damping Here and M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted

96. MOR and MCD` TD-DFT formulation with damping or Here M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted

97. MOR and MCD TD-DFT formulation with damping or M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted

98. MOR and MCD M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,submitted For m>n

99. MCD spectra of Porphyrins containing Mg,Ni and Zn

100. Example 2: Double Excitations Example 2: Double Excitations Application of the TRIC method Seth,M., Ziegler,T., J. Chem. Phys., 2005,123, 144105, Seth,M. ; Ziegler,T. J. Chem. Phys. 2006, 124, 144105

101. The B term : practical calculations The Calculation of the B -term

102. Limitations of Traditional TD-DFT What do we do with a degenerate ground state that can not be represented by a single Slater determinant ? What do we do with a degenerate ground state that can not be represented by a single Slater determinant ? Degenerate Ground State What are the fundamental equations ? What are the fundamental equations ? How do we calculate excitation energies How do we calculate excitation energies

103. Transformed Reference with an Intermediate Configuration Kohn Sham (TRICKS) TDDFT Solution: TRICKS of the Trade: Calculating the Excitation Energies of Molecules with Degenerate Ground States using TD-DFT Idea: Avoid problems with a degenerate ground state by taking an excited state that is nondegenerate as the (Transformed) Reference Intermediate Configuration. A. I. Krylov, Acc. Chem. Res. 2006, 39, 83-91

104. Example 2: d 1 transition metal complexes of T d symmetry, d-d transition Example 2: d 1 transition metal complexes of T d symmetry, d-d transition Application of the TRIC method

105. Result 2: d 1 transition metal complexes of T d symmetry, d-d transition Result 2: d 1 transition metal complexes of T d symmetry, d-d transition Application of the TRIC method

106. Example 3: d 1 transition metal complexes of T d symmetry, charge transfer Example 3: d 1 transition metal complexes of T d symmetry, charge transfer Application of the TRIC method

107. Result 3: d 1 transition metal complexes of T d symmetry, charge transfer Result 3: d 1 transition metal complexes of T d symmetry, charge transfer Application of the TRIC method

108. Conclusion Method for calculating the MCD A term (and dipole strength D ) within TD-DFT is outlined. Procedure for calculating C/D more straightforward. Implemented into the Amsterdam Density Functional Theory (ADF) program Applications to a range of small molecules Further information can be found in M. Seth, T Ziegler, A Banerjee, J. Autschbach, S.J.A. van Gisbergen E. J. Baerends, J. Chem. Phys. 120,10942, 2004 and M. Seth, T. Ziegler, J. Autschbach, J. Chem. Phys. accepted for publication.

109. MOR and MCD Consider a planar polarized light traveling a distance l through a media of randomly oriented molecules along the direction of a constant magnetic field with strength B. Consider a planar polarized light traveling a distance l through a media of randomly oriented molecules along the direction of a constant magnetic field with strength B. Here V(  ) is called the Verdet constant For such a system the plane of polarization will rotate by an angle  given by For such a system the plane of polarization will rotate by an angle  given by a M. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105, A.Banerjee,J.Autschbach,T.Ziegler Int.J.Quant.Chem.2006,101,572

110. MOR and MCD M. Krykunov, A. Banerjee, T. Ziegler,J. Autschbach J. Chem. Phys. 2005, 122, 075105,

111. MOR and MCD For m>n M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107

112. MOR and MCD M.Krykunov,M.Seth,T.Ziegler,J.Autschbach J.Chem.Phys. 2007,127,244107