Dipole moments.

For neutral systems, the dipole moment is calculated from the atomic charges and the lone-pairs as

$\begin{displaymath}\mu_x = cC\sum_AQ_Ax_A + cCa_o2\sum_A P(s-p_x)_AD_1(A) \end{displaymath}$

$\begin{displaymath}\mu_y = cC\sum_AQ_Ay_A + cCa_o2\sum_A P(s-p_y)_AD_1(A) \end{displaymath}$

$\begin{displaymath}\mu_z = cC\sum_AQ_Az_A + cCa_o2\sum_A P(s-p_z)_AD_1(A) \end{displaymath}$

$\begin{displaymath}\mu = \mu_x+\mu_y+\mu_z \end{displaymath}$

Where c = speed of light, C = charge on the electron, and a_o = Bohr radius, or cC = 2.99792458*1.60217733 = 4.8032066, and cCa₀2 = 2.99792458*1.60217733 = 4.8032066*0.529177249*2.0 = 5.0834948.. D₁(A) is defined elsewhere.

Conversion factors between Dipole units

Quantity	Dipole as	Factor (units)	Factor(value)	System
1 electron*1 Angstrom	Debye (D)	cC	4.80320	esu
1 Debye (D)	Dipole Length (m)	1/(cC)	2.0819*10^-11	SI

1 statvolt = 299.79 volts = c*10^-6 volts (c in m/s)

Formally, the dipole moment for an ion is undefined; however, it is convenient to set up a 'working definition.' Consider a heteronuclear diatomic ion in a uniform electric field. The ion will accelerate. To compensate for this, it is convenient to consider the ion in an accelerating frame of reference. The ion will experience a torque which acts about the center of mass, in a manner similar to that of a polar molecule. This allows us to define the dipole of an ion as the dipole the system would exhibit while accelerating in a uniform electric field. To formalize this definition:

$\begin{displaymath}\mu_x = cC\sum_AQ_A(x_A-x_{cog}) + cCa_o2\sum_A P(s-p_x)_AD(A) \end{displaymath}$

$\begin{displaymath}\mu_y = cC\sum_AQ_A(y_A-y_{cog}) + cCa_o2\sum_A P(s-p_y)_AD(A) \end{displaymath}$

$\begin{displaymath}\mu_z = cC\sum_AQ_A(z_A-z_{cog}) + cCa_o2\sum_A P(s-p_z)_AD(A) \end{displaymath}$

$\begin{displaymath}\mu = \mu_x+\mu_y+\mu_z, \end{displaymath}$

where x_cog is the x-coordinate of the center of gravity of the system

$\begin{displaymath}x_{cog} = \sum_AM_Ax_A, \end{displaymath}$

and y_cog and z_cog have similar definitions. This general expression will work for all discrete species, charged and uncharged, and is rotation and position invariant.