Molecular properties#

For static fields#

Within the appropriate limits of validity of perturbation theory, the total energy \(E\) of a molecule in static fields can be expanded in terms of multipole moments, defining static properties, as follows [Bar04]

\[\begin{equation*} E=E^{(0)} -\mu_\alpha E_\alpha -\frac{1}{2} \alpha_{\alpha \beta} E_\alpha E_\beta -\frac{1}{6} \beta_{\alpha \beta \gamma} E_\alpha E_\beta E_\gamma -\frac{1}{24} \gamma_{\alpha \beta \gamma \delta} E_\alpha E_\beta E_\gamma E_\delta +\cdots -\frac{1}{3} \Theta_{\alpha \beta} \nabla_\beta E_{\alpha} -\frac{1}{3} A_{\alpha , \beta \gamma} E_\alpha \nabla_\gamma E_{\beta} -\frac{1}{6} C_{\alpha \beta , \gamma \delta} \nabla_\beta E_{\alpha} \nabla_\delta E_{\gamma} +\cdots -\frac{1}{2} \xi_{\alpha \beta} B_\alpha B_\beta -\frac{1}{2} \chi_{\gamma , \alpha \beta} B_\alpha B_\beta E_\gamma -\frac{1}{4} \eta_{\gamma \delta ,\alpha \beta} B_\alpha B_\beta E_\gamma E_\delta +\cdots \end{equation*}\]

Einstein implicit summation over repeater indices is assumed. Above \(E^{(0)}\), a scalar, rank 0 tensor, is the energy in the absence of external fields; the electric (\(\mathbf{E}\), rank 1 tensor, polar, even time-reversality, indicated here in short as \(PE^1\)) and magnetic induction (\(\mathbf{B}\), rank 1 tensor, axial, odd time-reversality, indicated here as \(AO^1\)) fields have cartesian components \(E_\rho\) and \(B_\rho\), respectively (\(\rho = x,y,z\)), and the remaining elements of the equation above are (cartesian) tensor components of different rank, representing the molecular response to the applied fields. From the above equation it is evident that molecular multipoles in the static limit correspond to energy derivatives.



Tensor properties


induced electric dipole moment



electric dipole polarizability



electric dipole first hyperpolarizability



electric dipole second hyperpolarizability



induced electric quadrupole moment



electric dipole - quadrupole polarizability



electric quadrupole - quadrupole polarizability



induced magnetizability



electric first hypermagnetizability



electric second hypermagnetizability


The table just above classifies the lowest order molecular response moments in terms of their spatial symmetries. Time reversality classifies them with respect to time inversion and it is relevant only for frequency dependent fields/properties (the related tensor vanishes in the static limit). More below on the importance of this tensor classification of molecules multipoles.

For time-dependent fields#

When the molecule interacts with a radiation field of wavelength \(\lambda = \frac{2 \pi c_0}{\omega}\), \(c_0\) being the speed of light in vacuum, one can define the frequency-dependent properties by connecting them to the induced (in general also frequency-dependent) multipole moments. For a small harmonic perturbation of angular frequency \(\omega\) and keeping only linear terms in the perturbative expansion (only single photon processes) we can introduce the induced electric dipole moment \(\mathbf{\mu}\)

\[\begin{eqnarray*} \mu_\alpha & = & \alpha_{\alpha \beta}(-\omega;\omega) E_\beta^\omega + \frac{1}{\omega} G^\prime_{\alpha \beta}(-\omega;\omega)\dot{B}_\beta^\omega +\frac{1}{3} A_{\alpha, \beta \gamma}(-\omega;\omega) \nabla_\gamma E_{\beta} + \cdots \\ & & + \frac{1}{\omega} \alpha^\prime(-\omega;\omega) \dot{E}_\beta^\omega + G_{\alpha \beta}(-\omega;\omega) B_\beta^\omega + \frac{1}{3 \omega} A^\prime_{\alpha, \beta \gamma}(-\omega;\omega) \nabla_\gamma \dot{E}_{\beta}^\omega + \cdots \ ; \end{eqnarray*}\]

the induced electric quadrupole moment \(\mathbf{\Theta}\)

\[\begin{eqnarray*} \Theta_{\alpha \beta} & = & A_{\gamma,\alpha \beta}(-\omega;\omega) E_\gamma^\omega - \frac{1}{\omega} D^\prime_{\gamma,\alpha \beta}(-\omega;\omega)\dot{B}_\gamma^\omega + C_{\alpha \beta, \gamma \delta}(-\omega;\omega) \nabla_\delta E_{\gamma} + \cdots \\ & & - \frac{1}{\omega} A^\prime_{\gamma, \alpha \beta}(-\omega;\omega) \dot{E}_\gamma^\omega + D_{\gamma,\alpha \beta}(-\omega;\omega) B_\gamma + \frac{1}{\omega} C^\prime_{\alpha \beta,\gamma \delta} \nabla_\delta \dot{E}_{\gamma}^\omega + \cdots \ ; \end{eqnarray*}\]

and the induced magnetic dipole moment \(\mathbf{m}^\prime\)

\[\begin{eqnarray*} m_\alpha^\prime & = & \xi_{\alpha, \beta}(-\omega;\omega) B_\beta^\omega + \frac{1}{3 \omega} D^\prime_{\alpha,\beta \gamma}(-\omega;\omega)\nabla_\gamma \dot{E}_{\beta}^\omega - \frac{1}{\omega} G^\prime_{\beta \alpha}(-\omega;\omega) \dot{E}_{\beta} + \cdots \\ & & + \frac{1}{\omega} \xi^\prime_{\alpha \beta}(-\omega;\omega) \dot{B}_\beta^\omega + \frac{1}{3} D_{\alpha,\beta \gamma}(-\omega;\omega) \nabla_\gamma E_{\beta} + G_{\beta \alpha}(-\omega;\omega) E_{\beta}^\omega + \cdots \ . \end{eqnarray*}\]

Above we have made the dependence of the external fields on the angular frequency \(\omega\) explicit in the symbols \(E_\alpha^\omega\) and \(B_\alpha^\omega\). \(\dot{X} = \partial X / \partial t\), where \(t\) is the time variable, whereas \(\nabla_\alpha X_\beta\) indicates the spatial derivative along coordinate \(\alpha\) of the component \(\beta\) of the field \(\mathbf{X}\). As far as the molecular properties are concerned, they correspond to a generalization to the dynamical regime of the static properties defined in the expansion of the molecular energy above. In short, f.ex., \(\alpha_{\alpha \beta}\) in the energy expansion corresponds to \(\alpha_{\alpha \beta}(0;0)\) in the expansion of the induced electric dipole moment. The notation is chosen to specify the nature of the single photon interaction: a positive sign of the frequency \(\omega\) shows an incoming (absorbed) photon, whereas a negative sign indicates an outgoing (scattered, emitted) photon. In a general \(n\)-photon process the molecular response property \(X(-\omega_\sigma;\omega_n, \cdot , \omega_2, \omega_1)\) described a process implying the combination of absorption and emission of photons which, obeying to the law of energy conservation, leave the molecule in its stationary state and with \(\omega_\sigma = \omega_1+\omega_2+\cdots+\omega_n\). A convenient way of representing this process is through Ward’s graphs.

Tensors with a prime \(X^\prime\) vanish in the static limit (\(\omega \rightarrow 0\)), whereas tensors in the second rows of the three equation vanish for closed shell non-degenerate systems (described by real wavefunctions).

Response theory#

An equivalent (alternative) way to represent the induced multipole moments introduced just above is through the formalism of response theory. The second rank tensor representing the response to the perturbation represented by the one electron operator \(\hat{B}\) in the expansion of the induced moment represented by operator \(\hat{A}\) in our equations above is associated to linear response functions as follows

\[\begin{equation*} X(-\omega;\omega) \propto \langle \langle \hat{A} ; \hat{B} \rangle \rangle_{\omega} \end{equation*}\]

The operators \(\hat{A}\) and \(\hat{B}\) are better characterized in the following. Just as an example, the electric dipole polarizability is

\[\begin{equation*} \alpha(-\omega;\omega)_{\alpha \beta} = -\langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta \rangle \rangle_{\omega} \end{equation*}\]

This is because the perturbation to the electric dipole moment \(\mu\) by the electric field \(E_\omega\) is, in the dipole approximation, \(-{\mathbf{\mu}} \cdot {\mathbf{E}}\). The tensors introduced above can therefore be connected to the corresponding response functions as indicated in the following Table


Response function


\(\Re{\langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta \rangle \rangle_{\omega}}\)


\(\Im{\langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta \rangle \rangle_{\omega}}\)


\(\Re{\langle \langle \hat{\mu}_\alpha ; \hat{m}_\beta \rangle \rangle_{\omega}}\)


\(\Im{\langle \langle \hat{\mu}_\alpha ; \hat{m}_\beta \rangle \rangle_{\omega}}\)

\(A_{\alpha,\beta \gamma}\)

\(\Re{\langle \langle \hat{\mu}_\alpha ; \hat{\Theta}_{\beta \gamma} \rangle \rangle_{\omega}}\)

\(A^\prime_{\alpha,\beta \gamma}\)

\(\Im{\langle \langle \hat{\mu}_\alpha ; \hat{\Theta}_{\beta \gamma} \rangle \rangle_{\omega}}\)

\(C_{\alpha\beta,\gamma \delta}\)

\(\Re{\langle \langle \hat{\Theta}_{\alpha \beta} ; \hat{\Theta}_{\gamma \delta} \rangle \rangle_{\omega}}\)

\(C^\prime_{\alpha\beta,\gamma \delta}\)

\(\Im{\langle \langle \hat{\Theta}_{\alpha \beta} ; \hat{\Theta}_{\gamma \delta} \rangle \rangle_{\omega}}\)

\(D_{\alpha, \beta \gamma}\)

\(\Re{\langle \langle \hat{m}_{\alpha} ; \hat{\Theta}_{\beta \gamma} \rangle \rangle_{\omega}}\)

\(D^\prime_{\alpha, \beta \gamma}\)

\(\Im{\langle \langle \hat{m}_{\alpha} ; \hat{\Theta}_{\beta \gamma} \rangle \rangle_{\omega}}\)

\(\xi_{\alpha, \beta}\)

\(\Re{\langle \langle \hat{m}_{\alpha} ; \hat{m}_{\beta} \rangle \rangle_{\omega}} + \sum_i \frac{q_i^2}{4 m_i} \langle n \mid r_{i \alpha} r_{i \beta} - r_i^2 \delta_{\alpha \beta} \mid \rangle\)

\(\xi^\prime_{\alpha, \beta}\)

\(\Im{\langle \langle \hat{m}_{\alpha} ; \hat{m}_{\beta} \rangle \rangle_{\omega}}\)

In this section nevertheless we will be interested in nonlinear molecular response, and will discuss molecular properties of order higher than linear in most cases. Some of these appear, in their static (\(\omega_j = 0\)) form, in the perturbative expansion of the energy at the top of this section


Response function


\(\Re{\langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta, \hat{\mu}_\gamma \rangle \rangle_{\omega_1,\omega_2}}\)


\(\Re{\langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta, \hat{\mu}_\gamma, \hat{\mu}_\delta \rangle\rangle_{\omega_1,\omega_2,\omega_3}}\)


\(\Re{\langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta, \hat{m}_\gamma, \hat{m}_\delta \rangle \rangle_{\omega_1,\omega_2,\omega_3}}\)

The dependence on the angular frequencies above multiplies the number of possible molecular response properties as the order of the process increases. In the following we will connect these molecular response properties to the appropriate response functions and find and describe the connection to physical observables, showing how the tools of analytical response theory, that led in the last few decades to the development of highly efficient computational codes, allow us to confirm and in several case predict with increasing accuracy the outcome of experiments involving intense radiation impinging on molecules, aggregates, materials.

The operators involved in the definitions of the response functions are the electric dipole (\({\mu}\)), magnetic dipole (\({m}\)), traced electric quadrupole (\({q}\)) and diamagnetic susceptibility (\(\xi^{dia}\)) operators,

\[\begin{eqnarray*} {\mu}_\alpha & = & \sum_i q_i {r}_{i\alpha} \\ {m}_\alpha & = & \sum_i \frac{q_i}{2m_i} (r_i \times p_i)_\alpha = \sum_i \frac{q_i}{2m_i} \varepsilon_{\alpha\beta\gamma} {r}_{i\beta}{p}_{i\gamma} = \sum_i \frac{q_i}{2m_i} {l}_{i\alpha} \\ {q}_{\alpha\beta} & = & \sum_i q_i {r}_{i\alpha}{r}_{i\beta} \\ \xi_{\beta \gamma}^{dia} & = & \frac{1}{4} \sum_i \frac{q_i^2}{m_i}(r_{i \beta} r_{i \gamma} - r_{i \delta} r_{i \delta} \delta_{\beta\gamma}) \\ \mu_\alpha^p & = & \sum_i \frac{q_i}{m_i} p_{i\alpha} \end{eqnarray*}\]

with the standard notation for charges (\(q_i\)), masses (\(m_i\)) and cartesian components of the position (\({r}_{i\alpha}\)), angular momentum (\({l}_{i\alpha}\)) and linear momentum (\({p}_{i\alpha}\)) operators. We have included above also the definition of the dipole velocity operator (\(\mu^p\)). For one electron the operators simplify to (electrons, \(q_i = -1\), \(m_i = 1\))

\[\begin{eqnarray*} \mu_\alpha & = & -r_{\alpha} \\ m_\alpha & = & -\frac{1}{2} (r \times p)_\alpha = -\frac{1}{2} \epsilon_{\alpha\beta\gamma} r_\beta p_\gamma \\ q_{\alpha\beta} & = & -r_{\alpha} r_{\beta} \\ \xi_{\beta \gamma}^{dia} & = & \frac{1}{4} (r_{\beta} r_{\gamma} - r_{\delta} r_{\delta} \delta_{\beta\gamma}) \\ \mu_\alpha^p & = & -p_{\alpha} \end{eqnarray*}\]

Some other definitions (au)

\[\begin{eqnarray*} \mathcal{L}_\alpha & = &i [-(r \times \nabla)_\alpha] = i[-\epsilon_{\alpha\beta\gamma} r_\beta \nabla_\gamma] \\ m_\alpha & = & -\frac{\mathcal{L}_\alpha}{2} = -\frac{i}{2} [-(r \times \nabla)_\alpha] = -\frac{i}{2} [-\epsilon_{\alpha\beta\gamma} r_\beta \nabla_\gamma] \\ p_\alpha & = & -i \nabla_\alpha \ \ \ \ \ \Rightarrow \ \ \ \ \ \nabla_\alpha = i p_\alpha = -i \mu_\alpha^p \end{eqnarray*}\]

Tensor properties#

In this section wide use will be made of the concepts of space and time inversion symmetry in determining the form of the tensors introduced in the previous sections. Heavy reference will both explicitly and implicitly be made to Chapter 4 of the book by Laurence Barron [Bar04] and to the fundamental work of Birss on macroscopic symmetry in space-time cite(Birrs).

A given scalar, vector or tensor can be classified according to its behavior under space inversion and time reversal. Restricting our attention to vectors first, it is well known that they can be classified as polar or axial if they change (or do not change, respectively) their direction upon inversion of all coordinates (space inversion). Typical polar vectors are the position \(\vec{r}\) or the velocity vectors \(\vec{v}\) (and correspondingly the \({{\mathbf{\mu}}}\) and \({{\mathbf{\mu}}}^p\) operators), whereas a typical axial vector is the angular momentum vector \(\vec{l}\) (and the corresponding \({{\mathbf{m}}}\) operator). Vectors can also be classified on the basis of their behavior upon time reversal, i.e. a change reversing the arrow of time. Vectors that do not change their direction upon time reversal are said to be time even, whereas those changing their direction are said to be time odd. An example of a time even (or simply even vector is the position vector \(\vec{r}\), whereas both the velocity \(\vec{v}\) and the angular momentum \(\vec{l}\) operators are time odd. Particular examples of vectors whose symmetry characteristics upon space inversion and time reversal are of relevance here are the electric \(\mathbf{E}\) and magnetic induction \(\mathbf{B}\) fields (and their space and time derivatives). Their labels can be determined by examining the symmetry of the physical set-up generating them. If we indicate by \(P\) or \(A\) the condition of being a polar or axial vector, respectively, by \(E\) or \(O\) the condition of behaving as time-even or time-odd upon time reversality, and add a superscript \(1\) to refer to the particular rank of the vector as a special case of tensor (so that a superscript \(n\) indicates a tensor of rank \(n\), the symmetry relationships given in Table below (summarizing the spatial and time reversal symmetry properties of some relevant operators, fields and their time and space derivatives) can be obtained for the vectors we discussed in this paragraph.


Tensor properties









\(\mathbf{\nabla E}\)




\(\mathbf{\nabla \dot{E}}\)




\(\mathbf{\nabla B}\)




\(\mathbf{\nabla \dot{B}}\)


The definitions of polar/axial, even/odd can be extended to tensors of any rank. One way to determine the labels of a given tensor is to employ relationships that connect it to quantities whose symmetry behavior is known. Thus, f.ex., from the relationship

\[\begin{eqnarray*} \mu_\alpha & = & \alpha_{\alpha\beta} E_\beta \\ PE^1 & = & PE^2 \times PE^1 \end{eqnarray*}\]

we can determine that the electric dipole polarizability is a PE\(^2\) (polar, time-even, second rank) tensor. As a general rule, we can apparently use the following: the product of two polar or axial tensors yields a polar tensor, whereas the product of an axial with a polar tensor yields an axial tensor. The product of two even or of two odd tensors yield a time-even tensor, whereas the product of an even times an odd tensor yields a time-odd tensor. By further recognizing that the quadrupole tensor is polar even (PE\(^2\)), the use of Buckingham’s classical expressions for the real induced oscillating electric and magnetic multipole moments

\[\begin{eqnarray*} \mu_\alpha & = & \alpha_{\alpha\beta} E_\beta + \frac{1}{\omega} \alpha^\prime_{\alpha\beta} {\dot{E}}_\beta +\frac{1}{3} A_{\alpha\beta\gamma} {\nabla E}_{\beta\gamma} +\\ & & + \frac{1}{3\omega} A^\prime_{\alpha\beta\gamma} {\nabla \dot{E}}_{\beta\gamma} +G_{\alpha\beta} {B}_{\beta} + \frac{1}{\omega} G^\prime_{\alpha\beta} B_{\beta} + \cdots \\ \Theta_{\alpha\beta} & = & A_{\gamma\alpha\beta} {{E}}_{\gamma} - \frac{1}{\omega} A^\prime_{\gamma\alpha\beta} {{\dot{E}}}_{\gamma} + C_{\alpha\beta\gamma\delta} {{\nabla E}}_{\gamma\delta} + \frac{1}{\omega} C^\prime_{\alpha\beta\gamma\delta} {{\nabla\dot{ E}}}_{\gamma\delta} + \\ & & D_{\gamma\alpha\beta} {{B}}_\gamma -\frac{1}{\omega} D^\prime_{\gamma\alpha\beta} {{\dot{B}}}_\gamma + \cdots \\ m^\prime_{\alpha} & = & \xi_{\alpha\beta} {{B}}_{\beta} + \frac{1}{\omega} \xi^\prime_{\alpha\beta} {{\dot{B}}}_{\beta} + \frac{1}{3} D_{\alpha\beta\gamma} {{\nabla E}}_{\beta\gamma} + \frac{1}{3\omega} D^\prime_{\alpha\beta\gamma} {{\nabla\dot{E}}}_{\beta\gamma} + \\ & & G_{\beta\alpha} {{E}}_\beta -\frac{1}{\omega} G^\prime_{\beta\alpha} {{\dot{E}}}_\beta + \cdots \end{eqnarray*}\]

allows us to build the list in the following Table


Tensor properties

























A good rule of thumbs to determine the symmetry labels of the tensors consists in obtaining the label by taking the product of the labels of the individual operators, according to the prescription given just above. Thus f.ex., for the tensor \(\langle\langle{\mu}_\alpha; {m}_\beta,{\mu}^p_\alpha\rangle\rangle\), we take

\[\begin{eqnarray*} \langle\langle{\mu}_\alpha; {m}_\beta,{\mu}^p_\alpha\rangle\rangle \nonumber \\ {\mu}_\alpha \otimes {m}_\beta \otimes {\mu}^p_\alpha \nonumber \\ PE^1 \otimes AO^1 \otimes PO^1 \Rightarrow AE^3 \nonumber \end{eqnarray*}\]

With the information in our hand we can then resort to the Tables published by Birss and reproduced also in Barron’s book, to determine the tensors which survive for the different molecular symmetries. For example, for atoms, centrosymmetric linear molecules and etheronuclear diatomics, according to Birss, see Table 4 therein. For non centrosymmetric linear molecules (C\(_{\infty v}\), cf. magnetic point group 6mm) centrosymmetric linear molecules (D\(_{\infty h}\), cf. magnetic point group 6/mmm) and for atoms (S\(_3\), cf. magnetic point group m3m), the nonvanishing tensors are (\(n=0,1,2,\cdots\))

Point group

Tensor properties

C\(_{\infty v}\) (6mm)

All tensors

D\(_{\infty h}\) (6/mmm)

\(PE^{2n}\), \(AE^{2n+1}\), \(PO^{2n}\), \(AO^{2n+1}\)

S\(_3\) (m3m)

\(PE^{2n}\), \(AE^{2n+1}\), \(PO^{2n}\), \(AO^{2n+1}\)



Note that Birss’s Tables also specify the relationships between the various tensor components, and it is to these relationships that one can resort to introduce simplifications.

Electric properties#

Here we consider molecular properties which characterize the interactions with static and/or frequency-dependent electric fields. The electric properties of a molecule determine the electric properties of the bulk sample, such as the relative permittivity (dielectric constant) and the refractive index. In addition, the electric properties can be used to describe intermolecular forces.

Only static and dynamic molecular properties involving electric dipole and quadrupole operators will be discussed below.

Most theoretical results are obtained in atomic units (a.u.), whereas experimental data often are given in SI or esu units. In Here you find some useful conversion factors are given.

1 a.u.




\(E_h e^{-1}a_0^{-1}\)

5.104206 \(\times 10^{11}\) Vm\(^{-1}\)

1.7153 \(\times 10^{7}\) statvolt cm\(^{-1}\)


\(e a_0\)

8.478353 \( \times 10^{-30} \) C m

2.5417 \(\times 10^{-18}\)


\(e a_0^2\)

4.486551 \(\times 10^{-40}\) C m\(^2\)

1.3450 \( \times 10^{-26}\) statvolt cm\(^3\)


\(e^2 a_0^{2} E_h^{-1}\)

1.648777 \(\times 10^{-41}\) C\(^2\) m\(^2\) J\(^{-1}\)

1.4818 \( \times 10^{-25}\) & cm\(^{3}\)


\(e^3 a_0^{3} E_h^{-2}\)

3.206362 \( \times 10^{-53}\) C\(^3\) m\(^3\) J\(^{-2}\)

8.6392 \( \times 10^{-33}\) statvolt\(^{-1}\) cm\(^{4}\)


\(e^4 a_0^{4} E_h^{-3}\)

6.235381 \( \times 10^{-65}\) C\(^4\) m\(^4\) J\(^{-3}\)

5.0367 \( \times 10^{-40}\) statvolt\(^{-2}\) cm\(^{5}\)

The origin of the frequency-dependent electric dipole polarizability and hyperpolarizabilities#

The electric dipole moment \(\tilde{\mu}\) in the presence of an applied spatially homogeneous non-monochromatic electric field \( \sum_{i} \mathbf{F}^{\omega_{i}}\) (where \(\omega_{i}\) indicates the circular frequency) can be written as

\[\begin{eqnarray*} \tilde{\mu}_\alpha^{\omega_{\sigma}} & = & \mu_\alpha \delta(\omega_{\sigma}=0) + \alpha_{\alpha\beta}(-\omega_{\sigma};\omega_1)F_\beta(\omega_1) \\ & + & \frac{1}{2} \sum_{\omega_1,\omega_2} \beta_{\alpha\beta\gamma}(-\omega_{\sigma};\omega_1,\omega_2) F_\beta(\omega_1)F_\gamma(\omega_2) \\ & + & \frac{1}{6} \sum_{\omega_1,\omega_2,\omega_3} \gamma_{\alpha\beta\gamma\delta}(-\omega_{\sigma};\omega_1,\omega_2,\omega_3) F_\beta(\omega_1)F_\gamma(\omega_2)F_\delta(\omega_3) + \cdots, \end{eqnarray*}\]

where \(\mu_\alpha\) is the unperturbed electric dipole moment. In this phenomenological expansion (which we use to describe elastic light scattering processes) the summations over optical frequencies will be restricted in order to preserve a constant \(\omega_{\sigma}\) equal to the sum of optical frequencies

\[ \omega_{\sigma} = \sum_{i}\omega_i. \]

The equation above defines the components of the dynamic electric dipole polarizability tensor, \(\alpha_{\mu\nu}(-\omega_{\sigma};\omega_1)\), those of the first electric dipole hyperpolarizability, \(\beta_{\mu\nu\eta}(-\omega_{\sigma};\omega_1,\omega_2)\) and those of the second electric dipole hyperpolarizability \(\gamma_{\mu\nu\eta\xi}(-\omega_{\sigma};\omega_1,\omega_2,\omega_3)\). The combination of frequencies and the presence or absence of static fields characterize various nonlinear processes. The calculations can be done for any combination of frequencies satisfying the condition \(\omega_{\sigma} = \sum_{i}\omega_i\).

The Figure shows how some of these processes can be interpreted in terms of electronic transitions between ground and excited states. The excited states can be treated as virtual states, but if one of the excitation energies is close to the frequency of the light, the contribution of that state is often dominant. Simplified models that include in the analysis only a few excited states can be used when these computed states describe well the real, physical states of the molecule which are relevant for the property studied. However, it is important to emphasize that even though these few-states models in many cases may give qualitative insight into the electronic processes, the convergence of the sum-over-states expansion for the (hyper)polarizabilities
with respect to the number of excited states included in the summation is in general very slow.

We should also keep in mind that if an excitation energy equals a frequency, we have a very different situation in which we have to consider the finite lifetime of the excited state and also deal with the absorption process.

In experiment, the molecule is often perturbed by a combination of a static and a dynamic electric field. We then have, assuming a monochromatic dynamic field,

\[ F_\alpha(t) = F_\alpha^0 + F_\alpha^{\omega}\cos(\omega t), \]

and therefore the time dependence of the polarization can be described as

\[ \mu_\alpha(t) = \mu_\alpha^0 + \mu_\alpha^{\omega}\cos(\omega t) + \mu_\alpha^{2\omega}\cos(2\omega t)+\mu_\alpha^{3\omega}\cos(3\omega t) + \cdots, \]

where \(\mu_\alpha^{n\omega}\) are the Fourier amplitudes to be determined. In order to compare the measured polarization at a certain frequency to the theoretical expressions, we insert the expression of the dynamic field into the expression of the electric dipole moment \(\mu\) and use the relations between \(\cos^n(\omega t)\) and \(\cos(n\omega t)\). We obtain

\[\begin{eqnarray*} \mu^0_\alpha & = & \mu_\alpha + \alpha_{\alpha\beta} (0;0) F_\beta^0 + \frac{1}{2} \beta_{\alpha\beta\gamma}(0;0,0) F_\beta^0F_\gamma^0 + \frac{1}{6} \gamma_{\alpha\beta\gamma\delta}(0;0,0,0) F_\beta^0D_\gamma^0F_\delta^0 \\ & + & \frac{1}{4} \beta_{\alpha\beta\gamma}(0;-\omega,\omega) F_\beta^{\omega}F_\gamma^{\omega} + \frac{1}{4} \gamma_{\alpha\beta\gamma\delta}(0;0,-\omega,\omega) F_\beta^0F_\gamma^{\omega}F_\delta^{\omega}, \\ \mu^{\omega}_\alpha & = & \alpha_{\alpha\beta}(-\omega;\omega) F_\beta^{\omega} + \beta_{\alpha\beta\gamma}(-\omega;\omega,0) F_\beta^{\omega}F_\gamma^0 \\ & + & \frac{1}{2} \gamma_{\alpha\beta\gamma\delta}(-\omega;\omega,0,0)F_\beta^{\omega}F_\gamma^0F_\delta^0 + \frac{1}{8} \gamma_{\alpha\beta\gamma\delta}(-\omega;\omega,-\omega,\omega) F_\beta^{\omega}F_\gamma^{\omega}F_\delta^{\omega}, \\ \mu^{2\omega}_\alpha & = & \frac{1}{4} \beta_{\alpha\beta\gamma}(-2\omega;\omega,\omega) F_\beta^{\omega}F_\gamma^{\omega} + \frac{1}{4} \gamma_{\alpha\beta\gamma\delta}(-2\omega;\omega,\omega,0) F_\beta^{\omega}F_\gamma^{\omega}F_\delta^0, \\ \mu^{3\omega}_\alpha & = & \frac{1}{24} \gamma_{\alpha\beta\gamma\delta}(-3\omega;\omega,\omega,\omega) F_\beta^{\omega}F_\gamma^{\omega}F_\delta^{\omega}, \end{eqnarray*}\]

where we have truncated the expansions at \(\cos(3\omega t)\). Using the expansions given above and the expansion of \(\mu\) we can compare the experimentally measured polarization and the theoretical estimates.