# ORCA: Dipole moment of LiH from SP calculation

I calculate LiH with ORCA

%method
runtyp SP
end
%scf
hftyp RHF        # Selects closed-shell SCF
end
%basis
Basis "def2-QZVPP " # The orbital expansion basis set
end

%coords
CTyp xyz       # the type of coordinates = xyz or internal
Charge 0       # the total charge of the molecule
Mult 1         # the multiplicity = 2S+1 ; S = +1/2 - 1/2  = 1/2
Units Angs     # the unit of length = angs or bohrs
coords
Li        0.000000       0.00000        0.00000
H         1.594900       0.00000        0.00000
end
end
%output
Print[ P_Basis ] 2
Print[ P_MOs ] 1
end


Results is

---------------------------
INTERNAL COORDINATES (A.U.)
---------------------------
Li     0   0   0     0.000000000000     0.00000000     0.00000000
H      1   0   0     3.013924210991     0.00000000     0.00000000

-----------------------
MULLIKEN ATOMIC CHARGES
-----------------------
0 Li:    0.299997
1 H :   -0.299997
Sum of atomic charges:    0.0000000

-------------
DIPOLE MOMENT
-------------
X             Y             Z
Electronic contribution:     -3.84433       0.00000       0.00000
Nuclear contribution   :      1.48516       0.00000       0.00000
-----------------------------------------
Total Dipole Moment    :     -2.35917       0.00000       0.00000
-----------------------------------------
Magnitude (a.u.)       :      2.35917
Magnitude (Debye)      :      5.99653


Ok. I know the charges on atoms (by Mulliken) $$q = 0.299997$$ a.u.с., also I know the distance between atoms $$d = 3.013924210991$$ a.u. So, using classical formula for dipole momentum, I get $$\mu = q\cdot d = 0.299997 \cdot 3.013924210991 =0.9041 \text{ a.u.}$$

but ORCA calculations gives me $$\mu = 2.35917$$ a.u.

Why such a difference? Why are these charges on atoms needed if they cannot even be used to calculate the dipole moment?

• Well, Mulliken charges are the worse, look for which method is ORCA using to calculate the dipole moment.
– Camps
May 9, 2022 at 22:26
• Partial charges aren't real. The distance is the same no matter what partial charges you use, but every partial charge method will produce different partial charges, and hence, different dipole moments. I imagine orca is using the wavefunction, which is much better. I think most programs spit out Mulliken charges just because most programs spit out Mulliken charges, it doesn't mean we use them, or should. May 9, 2022 at 22:40
• @B.Kelly Why are these charges on atoms needed if they cannot even be used to calculate the dipole moment? May 10, 2022 at 6:08
• @Sergio The short answer is that (1) the reproduction of dipole moment by partial charges becomes better for multiatomic molecules; (2) it is possible to generate atomic charges that exactly reproduce the dipole moments, by first calculating the exact dipole moment and the constraining the charges to reproduce the dipole moment, but then the charges will be size-inconsistent, meaning that the charges of an isolated molecule A are changed when another molecule B is put into the system, even if A and B are infinitely far away from each other. May 10, 2022 at 7:34

As suggested by Tyberius, I now expand my comment into an answer.

The true charge density of a molecule is a sum of nuclear and electronic contributions. While the former is a sum of delta functions (under the Born-Oppenheimer approximation and neglecting the finite nuclear size effects), the latter has the form of an electronic cloud. Thus, the usual picture that Li bears some positive charge and H bears some negative charge is necessarily a simplification, since the electrons do not just reside on the Li and H centers, but are floating around them; moreover they do not float around the nuclei in a "symmetric" fashion, but will spend more time between the nuclei, compared to what would be expected if the electron density of LiH is simply a sum of two spherically symmetric densities.

The important corollary is that, no atomic charge can exactly reproduce the electric field of the LiH molecule, or essentially any molecule with more than one atom. With diatomic molecules, one is lucky enough that there is a choice of atomic charges that exactly reproduces the dipole moment (and the choice is unique). But this only means that they reproduce the leading order ($$R^{-3}$$) term of the electric field exactly; the higher order terms (quadrupole $$R^{-4}$$, octupole $$R^{-5}$$ etc.) are not necessarily calculated exactly or even accurately. As all chemistry happens at a finite distance from a molecule, one is rarely solely interested in reproducing the dipole moment alone. That is to say, yes, the dipole moment is more important than the quadrupole moments, even more than the octupole moments, etc., but one is usually happy to sacrifice a little accuracy in the dipole moment for a large improvement of the higher multipole moments. Thus, although it is usually a good thing to have the atomic charges reproduce the dipole moments approximately, there is basically no practical gain with reproducing them exactly.

Nevertheless, you can see that the dipole moment predicted by the Mulliken charges does not even approximately reproduce the correct dipole moment calculated at the same level of theory. Why? Well, this simply means that Mulliken charges are generally not very useful, as already mentioned in the link that Geoff posted in his comment. Nevertheless, quantum chemistry programs usually calculate them (plus the similarly unreliable Löwdin charges), because they are trivial to implement and extremely cheap to calculate, so that calculating them does more good than bad. Besides, one important application of atomic charges is to rationalize experimental observations (like chemical reactivity), in which case the atomic charges need not be quantitatively accurate; rather, they only need to change in the correct direction when you change the substituents, conformation, etc., of the molecule. Usually, almost all atomic charge models satisfy this requirement. For more quantitative studies that require e.g. plotting the logarithm of reaction rate constant against an atomic charge, there is a more stringent requirement that the atomic charge must be approximately a linear function of the "true" atomic charge with a positive slope, but Mulliken charges usually satisfy even this requirement. Nevertheless, more robust charges like Hirshfeld and NPA charges are not much more expensive than Mulliken charges, and it is generally recommended to use these charges even when Mulliken charges probably suffice, at the cost of merely a few seconds to minutes of additional computational time. For example, see my article which plotted the logarithms of polymerization rate constants against the NPA charges of the radical centers of the polymer chain radicals, and obtained a nice linear relation; one can thus predict the polymerization rates of novel monomers by an NPA calculation, instead of a much more expensive transition state search.

Finally, if your application specifically requires that the charges should reproduce the electric field of the molecule as closely as possible, you should use atomic charges that are obtained by fitting the molecular electrostatic potential, such as the RESP, MK or CHELPG charges (ORCA supports the last one, which should be explicitly requested by the "! CHELPG" keyword). This usually happens when you want to fit a force field. As I said before, these give charges that reproduce the dipole moment very accurately, but not exactly, as the methods also take care to approximately reproduce the higher multipole moments, which compromises the accuracy of the dipole moment.

Dipole moment in Quantum Mechanicl

I dare to expand the answer of wzkchem5, taking into account how I understood (with some math).

The dipole moment of molecular system (which is described by wave function $$\Phi$$) is an observabe, so in QM we can write: $$\begin{equation*} \vec{\mu} = \int \Phi^* \hat{\mu} \,\, \Phi \vec{r}. \end{equation*}$$ which leads to $$\begin{equation}\label{DM}\tag{DM} \vec{\mu} = \int\limits_V \vec{r} \left[\sum Z_A \delta(\vec{r} - \vec{R}_A) - \rho(x,y,x)\right] d\vec{r}, \end{equation}$$ or in nucleus - electrons separated form $$\begin{equation}\label{N-e}\tag{N - e} \vec{\mu} = \sum_a Z_a \vec{R}_a - \int \vec{r} \rho(x,y,x) d\vec{r}, \end{equation}$$ This formula from quantum mechanics is similar to classical one, where $$\rho (x,y,z)$$ --- electron charge density.

Calculation of dipole moment by RHF method

In RHF (with MO LCAO) approximation, the \eqref{DM} leads to $$\begin{equation}\label{DM LCAO} \tag{DM from LCAO} \vec{\mu} = \sum_A Z_A\vec{R}_A - 2 \sum_p\sum_q P_{pq}\int_V \vec{r} \chi_p^* \chi_q d\vec{r}, \end{equation}$$ where $$\{P_{pq}\}$$ -- density matrix, $$\chi_q$$ --- basis set functions.

Quantum chemistry software calculates this with the HF method using the formula \eqref{DM LCAO}.

As for the charges on the atoms.

From another side, an atomic partial charges $$q_A$$ are a central concept in general chemistry and widely used as mentioned wzkchem5. But, unfortunately, they do not correspond to a single well-defined quantum mechanical observable, so we have't operator $$\hat{q}_A$$ and as a concequence, their definition is quite arbitrary and they are not required to accurately reproduce the electrical characteristics of the molecule.

The calculation according to the formula (for diatomic molecule) $$\begin{equation}\label{QR}\tag{QR} \vec{\mu}_{q} = q_{Li} \vec{R_{Li}} - q_H \vec{R_{H}} = q \vec R_{Li-H}, \end{equation}$$ where $$R_{Li-H}$$ - the distence between atoms, takes into account only a part of the contribution to the dipole moment of the molecule associated with interatomic charge transfer. Let's show it.

Let's introduce $$\vec r = \vec R_{Li} + \vec r_{Li}$$ and substitute into the formula\eqref{N-e}

$$\begin{equation} \vec{\mu} = \left(Z_{Li} - \int \rho(x,y,x) d\vec{r} \right) \vec R_{Li} + \left(Z_{H} - \int \rho(x,y,x) d\vec{r} \right) \vec R_{H} - \int \vec (\vec r_{H} + \vec r_{Li}) \rho(x,y,x) d\vec{r} = q_{Li} \vec R_{Li} + q_{H} \vec R_{H} - \int (\vec r_{H} + \vec r_{Li}) \rho(x,y,x) d\vec{r} = q_{Li} \vec R_{Li - H} - \int (\vec r_{H} + \vec r_{Li}) \rho(x,y,x) d\vec{r} = \mu_{q} - \int (\vec r_{H} + \vec r_{Li}) \rho(x,y,x) d\vec{r}. \end{equation}$$

The last term $$- \int (\vec r_{H} + \vec r_{li}) \rho(x,y,x) d\vec{r}$$ takes into account the fact that the eletron density distribution of the bound atom is not spherically symmetric, and each atom acquires a dipole electric polarization, relative to its nucleus.

So, we can see, the $$\vec{\mu}_q = q_{Li} \vec R_{Li-H}$$ which is determined by the partial charge on the atom is just a part of total dipole moment of molecule $$\vec{\mu}$$.

There are many methods for determining partial charges. In paper The Atomic Partial Charges Arboretum: Trying to See the Forest for the Trees the classification of charges is given, depending on the methods of their determination:

• Class I charges are derived from experimentally measurable properties, e.g., from observed deformation densities, from the electronegativity equalization principle, or from, e.g., dipole moments of diatomic and small (usually highly symmetric) polyatomic molecules.
• Class II charges are extracted from the molecular orbitals (Mulliken, NPA,…) or the electron density (Bader QTAIM, Hirshfeld,…).
• Class III charges are extracted from the wave function or electron density by fitting a physical observable (e.g., the electrostatic potential) derived from it.
• Class IV charges are based on semiempirical adjustment of a well-defined Class II or Class III model to better reproduce one or more physical observables (e.g. the dipole moment).