Chirality: Physical Chemistry - American Chemical Society

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Chapter 5

Vibrational Transition Current Density: Visualizing the Origin of Vibrational Circular Dichroism and Infrared Intensities

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Teresa B . F r e e d m a n , E u n a h Lee, a n d T a i p i n g Z h a o Department of Chemistry, Syracuse University, Syracuse, N Y 13244-4100

Vibrational transition current density (TCD) maps and charge-weighted nuclear displacement vectors can be used to visualize the electronic and nuclear contributions to infrared absorption (IR) and vibrational circular dichroism (VCD) intensities. Vibrational T C D , a vector-field plot related to the integrand of the electronic contribution to the velocity-form electric dipole transition moment, shows the flow of electron density produced by nuclear motion. Examples are presented showing a variety of current patterns for modes of (2S,3S)-oxirane-d , (S)-methyl lactate, L­ -alanine, and (S)-methyl chloropropionate, which provide insight into the origin of the observed V C D and IR intensities. 2

Introduction Vibrational circular dichroism (VCD) is the differential absorbance of left and right circularly polarized infrared radiation by a chiral molecule during vibrational excitation (1). Infrared absorption intensity arises from vibrationally-generated linear oscillation of charge, producing an oscillating

© 2002 American Chemical Society

Hicks; Chirality: Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

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electric dipole moment (1). V C D intensity arises from vihrationally-generated angular oscillation of charge occurring about the direction of linear charge oscillation. These linear and angular charge oscillations result in oscillating electric and magnetic dipole moments, respectively, and both are required for non-zero V C D intensity. Present computational chemistry methodology yields calculated IR and V C D intensities that agree well with experiment (1,2). However, these scalar intensity values arise from integration over electronic and nuclear coordinates, and do not provide insight into the nature and origin of the charge oscillations responsible for the intensities. Instead, a way to directly calculate and view nuclear and electronic currents produced by the vibrational motion is required. Numerous molecular modeling programs incorporate routines to animate the nuclear vibrational motion, which provides some understanding of the nuclear contribution to V C D and IR intensities when the nuclear displacements are weighted by the nuclear charge. We have recently developed a method for calculating and viewing vihrationallygenerated electron density motion, termed vibrational transition current density (TCD) (3-6). We provide here a theoretical development for V C D and IR intensity that identifies the charge flow contributions, and demonstrate the relationship of vibrational T C D to IR and V C D intensities. Finally, we provide examples of charge-weighted nuclear displacement and T C D vector field maps for a variety of vibrational modes, and demonstrate how these maps provide an understanding of the origin of IR and V C D intensities.

Theoretical Background Infrared absorption intensity is proportional to the dipole strength, the absolute square of the electric dipole transition moment. As seen from eq 1, the position-form dipole strength, DC, for a harmonic fundamental transition for normal mode a in the electronic ground-state g, at angular frequency ω , depends on the derivative of the position-form electric dipole moment, μ , with respect to normal mode Q . Non-zero dipole strength thus requires linear oscillation of charge during the vibration (1). α

Γ

a

(1)

Vibrational circular dichroism intensity is proportional to the rotational or rotatory strength, the scalar product of the electric and magnetic dipole transition moments. The expression shown in eq 2 for the position-form

Hicks; Chirality: Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

67 rotational strength, R?, involves the derivative of the magnetic dipole moment, m, with respect to the conjugate vibrational momentum P . V C D intensity thus requires vibrationally-generated circular or angular oscillation of charge about the direction of linear charge oscillation (1). a

~2{dQ )

(2)

[dP )

a 0

a 0

The dipole and rotational strengths can also be expressed in terms of the velocity-form electric dipole moment, μ , in which case the electric dipole moment derivative is with respect to P . The velocity-form rotational strength is given by ν

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a

W - ^ ^ l ) = (a^Re(A ) .(m) =v

0 1

(3)

1 0

« /o

\ aJ dP

0

The Cartesian components of the electric and magnetic dipole transition moments are further expressed in terms of the atomic polar tensor (APT), P^ [position-form implied] or Ρ

Α ν αβ

[velocity-form], or the atomic axial tensor

(AAT), M^, for atom A, shown i n eqs 4-6. The electronic and nuclear contributions to the atomic polar tensor and atomic axial tensor elements are compiled i n Table 1. In these expressions, summation over Cartesian directions for repeated Greek subscripts is implied, R is nuclear position, Aa

R is nuclear velocity, Z e is nuclear charge, δ β is the Kronecker delta, and ε β is the rank 3 antisymmetric tensor (ε β = 0 i f any of αβγ are the same and ε β = 1 or -1 i f αβγ is an even or odd permutation of the order xyz, respectively), which is used in expressing vector cross product elements. A tilde denotes a complex quantity; the velocity-form electric dipole transition moment and the magnetic dipole transition moment are pure imaginary quantities, whereas the A P T and A A T elements are real. The s-vector component, s = (dR /dQ ) =(dR /dP ^, is expressed either in terms of nuclear displacement or velocity. Aa

A

α

γ

α

γ

α

α

Aaa

Aa

a Q

Aa

γ

a

\l/2

V

« /

A

Γί^Ί

2ω

α

(4)

J l a & /ο

In our expressions for the atomic polar and atomic axial tensors (Table 1), we use a parallel definition for both tensors in terms of a dipole moment

Hicks; Chirality: Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

68

(5)

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(6)

Table 1. Electronic and Nuclear Contributions to the A P T and A A T Tensor

Electronic Contribution

Nuclear Contribution

Atomic Polar Tensor (APT) Position Form Μα

/Λ=0

=