Table 1.

The effective susceptibility ( ${\chi }_{\mathit{eff}}^{(2)}$) for each polarization combination in SFG includes a unique subset of the probed susceptibility tensor elements ( ${\chi }_{\mathit{ijk}}^{(2)}$) of the sample, many of which vanish for common sample symmetries. The non-zero elements often have interdependencies, however, note that the evaluation of the achiral and chiral tensor elements in this table assumes that a symmetry plane is aligned with the plane of incidence.

${\chi }_{\mathit{eff}}^{(2)}$	ijk of probed ${\chi }_{\mathit{ijk}}^{(2)}$	ijk of the non-vanishing ${\chi }_{\mathit{ijk}}^{(2)}$ depending on the sample symmetry
		symmetry	achiral elements	chiral elements
${\chi }_{\mathit{ppp}}^{(2)}$	zzz, zxx, xzx, xxz xxx, xzz, zxz, zzx	C1	All tensor elements are nonvanishing and independent.
${\chi }_{\mathit{ssp}}^{(2)}$	yyz, yyx	C2	zzz, zxx, xzx, xxz, yyz, yzy, zyy	xzy, zxy, xyz, zyx, yxz, yzx
${\chi }_{\mathit{sps}}^{(2)}$	yzy, yxy	C2v	zzz, zxx xzx, xxz, yyz, yzy, zyy	–
${\chi }_{\mathit{pss}}^{(2)}$	zyy, xyy	C3	xxy = xyx = yxx = −yyy, xxz = yyz xyy = yxy = yyx = −xxx, xzx = yzy zxx = zyy, zzz	xzy = −yzx, zxy = −zyx, xyz = −yxz
${\chi }_{\mathit{sss}}^{(2)}$	yyy	C3v	xyy = yxy = yyx = −xxx, xxz = yyz xzx = yzy, zxx = zyy, zzz	–
${\chi }_{\mathit{pps}}^{(2)}$	xzy, zxy, zzy, xxy	C4; C6; C∞	zzz, zxx = zyy, xzx = yzy, xxz = yyz	xzy = −yzx, zxy = −zyx, xyz = −yxz
${\chi }_{\mathit{psp}}^{(2)}$	xyz, zyx, zyz, xyx	C4v; C6v; C∞v	zzz, zxx = zyy, xzx = yzy, xxz = yyz	–
${\chi }_{\mathit{spp}}^{(2)}$	yxz, yzx, yzz, yxx	D∞	–	xzy = −yzx, zxy = −zyx, xyz = −yxz