Quadrupole Contributions from Interface and Bulk

Abstract

This chapter provides the comprehensive argument on quadrupole contributions in SFG spectroscopy. Though the preceding chapters have dealt with the induced dipole contribution, the accurate theory of SFG and SHG should take account of the induced quadrupole beside the dipole. The former can arise from the bulk region, while the latter stems only from the interface. We clarify three kinds of quadrupole contributions, namely χ ^IQ, χ ^IQB and χ ^B, in addition to the dipole one χ ^ID. The four terms have different roles in SFG and SHG spectra, as summarized in Sect. 7.5. The following discussion elucidates these quadrupole mechanisms and characters, and derives their microscopic formulas for calculating them.

Appendices

Appendix

A.1 Physical Meaning of χ ^IQB

Among the four terms of nonlinear susceptibility summarized in Fig. 7.5, χ ^IQB may be difficult to understand intuitively. It arises from the integral of nonlinear polarization over the interface region in Sect. 7.2.3, though it is a pure bulk property. The χ ^IQB term is known to have significant contribution in some SFG spectra [12, 13]. Here we explain its origin and physical meaning in an illustrative manner.⁶

As we discussed in Sect. 7.2.3, the χ ^IQB term in Eq. (7.30) originates from the integral of quadrupole over the interface. Accordingly, we consider the situation in Fig. 7.6 that the induced quadrupole is distributed over the whole interface system, and evaluate the net polarization of interface P ^I in Eq. (7.26) by integrating the dipole polarization over the interface region from z = z _b to ∞. The lower bound z = z _b is arbitrarily chosen in the bulk region so that the integral encompasses the whole interface region. In this integral of polarization, the distributed quadrupole comes into play at the lower bound z = z _b. The molecules located across the threshold z = z _b partially contribute to the integral, since the divided quadrupole moments Q _yz and Q _zz bring net dipole moments μ _y and μ _z, respectively, in the integral of P ^I. The contribution necessarily arises from the uniformly distributed quadrupole, irrespective of the location of the threshold z = z _b. We note that this mechanism is essentially common to the role of quadrupole on the surface potential [22].

Fig. 7.6

(Left) Schematic picture of the quadrupole (Q _zz) distribution in the interface system, where the lower bound of the integral z = z _b is shown by red line. (Right) Divided quadrupole moments (Q _yz, Q _zz) by the threshold z = z _b bring net dipole moments (μ _y, μ _z) in the integral region z > z _b

The integral of quadrupole contributions could be understood without resorting to the arbitrary threshold of z _b. This mechanism of χ ^IQB is related to the infinite summation of oscillating terms. A quadrupole is regarded as a pair of antiparallel dipoles, as illustrated in Fig. 7.6. Thus the sum of all quadrupole contributions becomes equivalent to the infinite summation of a pair of antiparallel dipoles, (μ − μ) + (μ − μ) + (μ − μ) + ⋯, where each pair (μ − μ) corresponds to a quadrupole moment. This infinite summation could be defined on the basis of Abel summability [9],

$$\displaystyle \begin{aligned} (\mu - \mu) + (\mu - \mu) + \cdots = \sum_{n=0}^\infty \mu \: (-1)^n = \lim_{x \to -1+0} \frac{\mu}{1-x} = \frac{\mu}{2},\end{aligned} $$

which yields a net dipole contribution. The above definition of this infinite sum can be obtained by using a proper convergence factor. Such oscillating sum elucidates the χ ^IQB contribution to the net dipole P ^I. We will encounter the analogous mechanism in the NaOH aqueous solution surface in Sect. 9.3.

A.2 Levi-Civita Antisymmetric Tensor

The Levi-Civita antisymmetric tensor ε _ijk is defined as follows.

$$\displaystyle \begin{aligned} {{\varepsilon }_{ijk}}=\left\{ \begin{array}{*{35}{l}} 1 & \left( ijk= xyz,yzx,zxy \right), \\ -1 & \left( ijk= yxz,zyx,xzy \right), \\ 0 & \left( \text{otherwise} \right). \\ \end{array} \right. {}\end{aligned} $$

(7.132)

This symbol is quite convenient when manipulating various formulas in the vector analysis.

Using the Levi-Civita symbols thus defined, the vector product is represented in the following form,

$$\displaystyle \begin{aligned} \left[ {\boldsymbol{B}} \times {\boldsymbol{C}} \right]_i = \varepsilon _{ijk} B_j C_k. \end{aligned}$$

In this section of Appendix we employ the Einstein’s convention of contraction, and omit the summation symbol $\sum$ over i, j, or k.⁷ The rotation of a vector is represented in a similar way, $\left [ \nabla \times {\boldsymbol {C}} \right ]_i = \varepsilon _{ijk} \partial _j C_k$. The scalar triple product is given by ${\boldsymbol {A}} \cdot \left ( {\boldsymbol {B}} \times {\boldsymbol {C}} \right ) = \varepsilon _{ijk} A_i B_j C_k$.

The Levi-Civita tensor satisfies the following contraction formulas,

$$\displaystyle \begin{aligned} \varepsilon _{ijk} \varepsilon _{pqk} &= \delta _{ip} \delta _{jq} - \delta _{iq} \delta _{jp}, {} \end{aligned} $$

(7.133)

$$\displaystyle \begin{aligned} \varepsilon _{ijk} \varepsilon _{pjk} &= 2 \delta _{ip}, {} \end{aligned} $$

(7.134)

$$\displaystyle \begin{aligned} \varepsilon _{ijk} \varepsilon _{ijk} &= 6. {} \end{aligned} $$

(7.135)

Proof

ε _ijk in Eq. (7.132) is expressed using a set of orthonormal vectors {e _x, e _y, e _z} as

$$\displaystyle \begin{aligned} \varepsilon_{ijk} = \left| \begin{matrix} ({\boldsymbol{e}}_x \cdot {\boldsymbol{e}}_i ) & ({\boldsymbol{e}}_x \cdot {\boldsymbol{e}}_j ) & ({\boldsymbol{e}}_x \cdot {\boldsymbol{e}}_k ) \\ ({\boldsymbol{e}}_y \cdot {\boldsymbol{e}}_i ) & ({\boldsymbol{e}}_y \cdot {\boldsymbol{e}}_j ) & ({\boldsymbol{e}}_y \cdot {\boldsymbol{e}}_k ) \\ ({\boldsymbol{e}}_z \cdot {\boldsymbol{e}}_i ) & ({\boldsymbol{e}}_z \cdot {\boldsymbol{e}}_j ) & ({\boldsymbol{e}}_z \cdot {\boldsymbol{e}}_k ) \end{matrix} \right| \equiv \left| \; {\boldsymbol{U}}(xyz, ijk) \; \right| . {} \end{aligned} $$

(7.136)

Therefore,

$$\displaystyle \begin{aligned} \varepsilon_{ijk} \varepsilon_{pqr} &= \left| \; {\boldsymbol{U}}(xyz, ijk) \; \right| \cdot \left| \; {\boldsymbol{U}}(xyz,pqr) \; \right| = \left| \; {\boldsymbol{U}}(xyz, ijk)^T \; \right| \cdot \left| \; {\boldsymbol{U}}(xyz, pqr) \; \right|\\ & = \left|\; {\boldsymbol{U}}(ijk, pqr) \; \right| \notag \\ &= \left| \begin{matrix} \delta_{ip} & \delta_{iq} & \delta_{ir} \\ \delta_{jp} & \delta_{jq} & \delta_{jr} \\ \delta_{kp} & \delta_{kq} & \delta_{kr} \end{matrix} \right| \notag \\ &= \delta_{ir} (\delta_{jp} \delta_{kq} - \delta_{jq} \delta_{kp}) - \delta_{jr} (\delta _{ip} \delta_{kq} - \delta_{iq} \delta_{kp}) + \delta_{kr} (\delta_{ip} \delta_{jq} - \delta_{iq} \delta_{jp}) . \end{aligned} $$

Equations (7.133), (7.134), (7.135) are proved by taking the contraction.

$$\displaystyle \begin{aligned} \varepsilon_{ijk} \varepsilon_{pqk} &= \delta_{ik} (\delta_{jp} \delta_{kq} - \delta_{jq} \delta_{kp}) - \delta_{jk} (\delta _{ip} \delta_{kq} - \delta_{iq} \delta_{kp}) + \delta_{kk} (\delta_{ip} \delta_{jq} - \delta_{iq} \delta_{jp}) \notag \\ &= \delta_{iq} \delta_{jp} - \delta_{ip} \delta_{jq} - \delta_{jq} \delta_{ip} + \delta_{jp} \delta_{iq} + 3 (\delta_{ip} \delta_{jq} - \delta_{iq} \delta_{jp}) \notag \\ &= \delta_{ip} \delta_{jq} - \delta_{iq} \delta_{jp} ,\end{aligned} $$

(7.133)

$$\displaystyle \begin{aligned} \varepsilon_{ijk} \varepsilon_{pjk} &= \delta_{ip} \delta_{jj} - \delta_{ij} \delta_{jp} = 3 \delta_{ip} - \delta{ip} = 2 \delta_{ip} ,\end{aligned} $$

(7.134)

$$\displaystyle \begin{aligned} \varepsilon_{ijk} \varepsilon_{ijk} &= 2 \delta_{ii} = 6 .\end{aligned} $$

(7.135)

□

These Eqs. (7.133), (7.134), (7.135), particularly Eq. (7.133), are extensively utilized in manipulating vector formulas. For example, the vector triple product is rearranged as follows,

$$\displaystyle \begin{aligned} & [ {\boldsymbol{A}} \times ( {\boldsymbol{B}} \times {\boldsymbol{C}} ) ]_i = \varepsilon _{ijk} A_j [ B \times C ]_k = \varepsilon _{ijk} A_j ( \varepsilon _{klm} B_l C_m ) \\ & = \varepsilon _{ijk} \varepsilon _{lmk} A_j B_l C_m = ( \delta _{il} \delta _{jm} - \delta _{im} \delta _{jl} ) A_j B_l C_m = A_j B_i C_j - A_j B_j C_i \\ & =B_i ( {\boldsymbol{A}} \cdot {\boldsymbol{C}}) - C_i ({\boldsymbol{A}} \cdot {\boldsymbol{B}}), \end{aligned} $$

where we have employed the permutation relation, ε _klm = ε _lmk, and Eq. (7.133) in the above derivation.

________________________________________________________________________________________________ [Problem 7.4] Derive the following formulas (i)–(vi) using the Levi-Civita tensor (see Sect. 7.6.4). (A, B, C, D refer to vectors, and ϕ to a scalar.)

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} &\text{(i) } && \left( {\boldsymbol{A}}\times {\boldsymbol{B}} \right)\cdot \left( {\boldsymbol{C}}\times {\boldsymbol{D}} \right) =\left( {\boldsymbol{A}}\cdot {\boldsymbol{C}} \right)\left( {\boldsymbol{B}}\cdot {\boldsymbol{D}} \right)-\left( {\boldsymbol{A}}\cdot {\boldsymbol{D}} \right)\left( {\boldsymbol{B}}\cdot {\boldsymbol{C}} \right) \\ &\text{(ii) } && \nabla \times \left( \nabla \times {\boldsymbol{A}} \right)=\nabla \left( \nabla \cdot {\boldsymbol{A}} \right)-{{\nabla }^{2}}{\boldsymbol{A}} \\ &\text{(iii) } && \nabla \cdot \left( \nabla \times {\boldsymbol{A}} \right)=0 \\ &\text{(iv) } && \nabla \times \left( \nabla \phi \right)=0 \\ &\text{(v) } && \nabla \cdot \left( {\boldsymbol{A}}\times {\boldsymbol{B}} \right) = {\boldsymbol{B}}\cdot \left( \nabla \times {\boldsymbol{A}} \right)-{\boldsymbol{A}}\cdot \left( \nabla \times {\boldsymbol{B}} \right) \\ &\text{(vi) } && \nabla \times \left( {\boldsymbol{A}} \times {\boldsymbol{B}} \right) = {\boldsymbol{A}} \left(\nabla \cdot {\boldsymbol{B}} \right) + \left( {\boldsymbol{B}} \cdot \nabla \right) {\boldsymbol{A}} - {\boldsymbol{B}} \left( \nabla \cdot {\boldsymbol{A}} \right) - \left( {\boldsymbol{A}} \cdot \nabla \right) {\boldsymbol{B}} \end{array}\end{aligned} $$

__________________________________________________________________________________

The Levi-Civita tensor is invariant under rotation of the coordinates,

$$\displaystyle \begin{aligned} \mathcal{D}_{ip} \mathcal{D}_{jq} \mathcal{D}_{kr} \: \varepsilon_{pqr} = \varepsilon_{ijk}, {} \end{aligned} $$

(7.137)

where $\mathcal {D}$ is the rotational matrix in Eq. (3.43). This feature will be utilized in Chap. 8.

Proof

Consider a rotational matrix ${\boldsymbol {\mathcal {D}}}$ that converts a set of orthonormal vectors {e _x, e _y, e _z} to $\{ {\boldsymbol {e}}_{x^\prime }, {\boldsymbol {e}}_{y^\prime }, {\boldsymbol {e}}_{z^\prime } \}$,

$$\displaystyle \begin{aligned} \begin{pmatrix} {\boldsymbol{e}}_{x^\prime} \\ {\boldsymbol{e}}_{y^\prime} \\ {\boldsymbol{e}}_{z^\prime} \end{pmatrix} = {\boldsymbol{\mathcal{D}}} \begin{pmatrix} {\boldsymbol{e}}_x \\ {\boldsymbol{e}}_y \\ {\boldsymbol{e}}_z \end{pmatrix}. \end{aligned}$$

Then the following relation is proved using the unitary character of ${\boldsymbol {\mathcal {D}}}$ ($\left | {\boldsymbol {\mathcal {D}}} \right | = 1$) and Eq. (7.136),

$$\displaystyle \begin{aligned} \varepsilon_{ijk} = \left| \; {\boldsymbol{U}} (xyz, ijk) \; \right| = \left| {\boldsymbol{\mathcal{D}}}^T \; {\boldsymbol{U}} (x^\prime y^\prime z^\prime, ijk) \; \right| = \left| \; {\boldsymbol{U}} (x^\prime y^\prime z^\prime, ijk) \; \right|. \end{aligned}$$

Suppose that pqr denote x ^′, y ^′, z ^′ while ijk denote x, y, z, the following equation is derived.

$$\displaystyle \begin{aligned} \mathcal{D}_{ip} \mathcal{D}_{jq} \mathcal{D}_{kr} \varepsilon_{pqr} = \mathcal{D}_{ip} \mathcal{D}_{jq} \mathcal{D}_{kr} \left| \; {\boldsymbol{U}} (x^\prime y^\prime z^\prime, pqr) \; \right| = \left| \; {\boldsymbol{U}} (x^\prime y^\prime z^\prime, ijk) \; \right|\notag \\ = \left| \; {\boldsymbol{U}} (xyz, ijk) \; \right| = \varepsilon_{ijk}.\end{aligned} $$

(7.137)

□

A.3 Definition of Bulk Polarization

In Chap. 7 the bulk polarization ${\boldsymbol {P}}^{\text{B}}_G$ is defined by Eq. (7.36),

$$\displaystyle \begin{aligned} &P_{G,p}^{\mathrm{B}} = i \int_{-\infty }^0 dz\sum_{q,r,s} \left\{ \chi _{pqrs}^{\mathrm{D1},\beta } (\Omega, \omega_1, \omega_2) k_{T,s}^\beta (\omega_1) + \chi _{pqrs}^{\mathrm{D2},\beta} (\Omega, \omega_1, \omega_2) k_{T,s}^\beta (\omega_2) \right. \notag \\ & \left. \qquad -\chi _{pqrs}^{\mathrm{Q},\beta} (\Omega, \omega_1, \omega_2) \left( k_{T,s}^\beta (\omega_1) + k_{T,s}^\beta (\omega_2) \right) \right\} \notag \\ & \qquad \cdot f_p^\beta (\Omega) f_q^\beta (\omega_1) f_r^\beta (\omega_2) L_{I,q} (\omega_1) L_{I,r} (\omega_2) E_{I,q}^\alpha (\omega_1) E_{I,r}^\alpha (\omega_2) \notag \\ & \qquad \cdot \exp \left[ i \left( k_{T,z}^\beta (\omega_1) + k_{T,z}^\beta (\omega_2) - k_{G,z}^\beta (\Omega) \right) z \right],\end{aligned} $$

(7.36)

which is consistent to Refs. [21] and [17], while a slightly different expression ${\boldsymbol {P}}{{ }_G^{\text{B}}}^\prime$ is found in Refs. [11] and [5]. In this Appendix, we clarify the difference between the two expressions.

Some other literature [5, 11] employs a different expression, ${\boldsymbol {P}}{{ }_{G}^{\mathrm {B}}}^\prime$, for the bulk polarization,

$$\displaystyle \begin{aligned} & P{{}_{G,p}^{\mathrm{B}}}^\prime = i \int_{-\infty }^0 dz \sum_{q,r,s} \left\{ \chi _{pqrs}^{\mathrm{D1},\beta } (\Omega, \omega_1, \omega_2) k_{T,s}^{\beta }(\omega_1) + \chi _{pqrs}^{\mathrm{D2},\beta }(\Omega, \omega_1, \omega_2) k_{T,s}^{\beta }(\omega_2) \right. \notag \\ & \left. \qquad -\chi _{pqrs}^{\mathrm{Q},\beta }(\Omega, \omega_1, \omega_2) k_{G,s}^{\beta }(\Omega ) \right\} \notag \\ & \qquad \cdot f_p^\beta (\Omega ) f_q^\beta (\omega_1) f_r^\beta (\omega_2) L_{I,q} (\omega_1) L_{I,r} (\omega_2) E_{I,q}^\alpha (\omega_1) E_{I,r}^\alpha (\omega_2) \notag \\ & \qquad \cdot \exp \left[ i \left( k_{T,z}^\beta (\omega_1) + k_{T,z}^\beta (\omega_2) - k_{G,z}^\beta (\Omega ) \right) z \right]. {} \end{aligned} $$

(7.138)

We notice a slight but significant difference; the third term of the integrand in Eq. (7.36) is $-\chi _{pqrs}^{Q,\beta } (\Omega , \omega _1, \omega _2) \left ( k_{T,s}^\beta (\omega _1) + k_{T,s}^\beta (\omega _2) \right )$, while that in Eq. (7.138) is $-\chi _{pqrs}^{Q,\beta } (\Omega , \omega _1, \omega _2) k_{G,s}^\beta (\Omega )$.

We could interpret that ${\boldsymbol {P}}^{\mathrm {B} \prime }_G$ in Eq. (7.138) involves both the bulk polarization ${\boldsymbol {P}}^{\mathrm {B}}_G$ and a part of interfacial polarization attributed to χ ^IQB. Equation (7.28) shows that the part of interfacial polarization P ^I attributed to χ ^IQB is

$$\displaystyle \begin{aligned} P_p^{\mathrm{IQB}} &= \sum_{q,r} \chi _{pqr}^{\mathrm{IQB}} (\Omega, \omega_1, \omega_2) L_{I,q} (\omega_1) L_{I,r}(\omega_2) E_{I,q}^\alpha (\omega_1) E_{I,r}^\alpha (\omega_2) \notag \\ & = \sum_{q,r} \chi _{pqrz}^{\mathrm{Q},\beta } (\Omega, \omega_1, \omega_2) f_p^\beta (\Omega ) f_q^\beta (\omega_1) f_r^\beta (\omega_2) L_{I,q} (\omega_1) L_{I,r} (\omega_2) E_{I,q}^\alpha (\omega_1) E_{I,r}^\alpha (\omega_2) \notag \\ & = i \int_{-\infty }^{0} dz\ \sum_{q,r,s}{\ } \delta_{sz} \left( k_{T,s}^\beta (\omega_1) + k_{T,s}^\beta (\omega_2) -k_{G,s}^\beta (\Omega ) \right) \chi _{pqrz}^{Q,\beta } (\Omega, \omega_1, \omega_2) \notag \\ & \qquad \cdot f_p^\beta (\Omega ) f_q^\beta (\omega_1) f_r^\beta (\omega_2) L_{I,q} (\omega_1) L_{I,r} (\omega_2) E_{I,q}^\alpha (\omega_1) E_{I,r}^\alpha (\omega_2) \notag \\ & \qquad \cdot \exp \left[ i \left( k_{T,z}^\beta (\omega_1) + k_{T,z}^\beta (\omega_2) - k_{G,z}^\beta (\Omega ) \right) z \right], {} \end{aligned} $$

(7.139)

where the form of χ ^IQB in Eq. (7.30) has been adopted in the above formula. Therefore, we can readily see the following relation,

$$\displaystyle \begin{aligned} P_{G,p}^{\mathrm{B}} + P_p^{\mathrm{IQB}} = P{{}_{G,p}^{\mathrm{B}}}^\prime . {} \end{aligned} $$

(7.140)

Although either definition for the bulk polarization could be used in principle, we recommend ${\boldsymbol {P}}_G^{\mathrm {B}}$ in Eq. (7.36) to describe the bulk polarization for the following reasons. First, ${\boldsymbol {P}}_G^{\mathrm {B}}$ is a well defined quantity with respect to the choice of origin as described in Sect. 7.4, and can be separately detected from the other terms by experimental measurements. Second, ${\boldsymbol {P}}_G^{\mathrm {B}}$ allows for distinguishing χ ^IQB and χ ^B. We have argued in Chap. 7 and Figure 7.5 that χ ^IQB and χ ^B have different physical meanings and different dependence on the optical geometry. The role of χ ^IQB has been increasingly recognized in the SFG spectroscopy [12, 13]. The former definition is convenient to examine the effects of χ ^IQB and χ ^B separately.