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| '''Mulliken charges''' arise from the '''Mulliken population analysis'''<ref>{{cite doi | 10.1063/1.1740588 }}</ref><ref>I. G. Csizmadia, Theory and Practice of MO Calculations on Organic Molecules, Elsevier, Amsterdam, 1976.</ref> and provide a means of estimating [[Partial charge|partial atomic charges]] from calculations carried out by the methods of [[computational chemistry]], particularly those based on the [[linear combination of atomic orbitals molecular orbital method]], and are routinely used as variables in linear regression (QSAR<ref name="isbn0-582-38210-6">{{cite book | author = Leach, Andrew R. | title = Molecular modelling: principles and applications | publisher = Prentice Hall | location = Englewood Cliffs, N.J | year = 2001 | isbn = 0-582-38210-6 }}</ref>) procedures.<ref>{{cite journal | last = Ohlinger | first = William S. | coauthors = Philip E. Klunzinger, Bernard J. Deppmeier, and Warren J. Hehre | title = Efficient Calculation of Heats of Formation | journal = The Journal of Physical Chemistry A | volume = 113 | issue = 10 | pages = 2165–2175 | publisher = ACS Publications | date = January 2009 | doi = 10.1021/jp810144q | pmid=19222177}}</ref> The method was developed by [[Robert S. Mulliken]], after whom the method is named. If the coefficients of the [[basis set (chemistry)|basis functions]] in the molecular orbital are '''C'''<sub>μi</sub> for the μ'th basis function in the i'th molecular orbital, the density matrix terms are:
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| :<math> \mathbf {D_{\mu\nu}} = \mathbf{2}\sum_{i} \mathbf {C_{\mu i}} \mathbf {C_{\nu i}^*} </math>
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| for a closed shell system where each molecular orbital is doubly occupied. The population matrix <math> \mathbf{P} </math> then has terms
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| :<math> \mathbf {P_{\mu\nu}} = \mathbf {D_{\mu\nu}} \mathbf {S_{\mu\nu}} </math>
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| <math> \mathbf{S} </math> is the overlap matrix of the basis functions. The sum of all terms of <math> \mathbf {P_{\nu\mu}} </math> summed over <math> \mathbf {\mu} </math> is the gross orbital product for orbital <math> \mathbf {\nu} </math> - <math> \mathbf {GOP_{\nu}} </math>. The sum of the gross orbital products is '''N''' - the total number of electrons. The Mulliken population assigns an electronic charge to a given atom '''A''', known as the gross atom population: <math> \mathbf {GAP_{A}} </math> as the sum of <math> \mathbf {GOP_{\nu}} </math> over all orbitals <math> \mathbf {\nu} </math> belonging to atom A. The charge, <math> \mathbf {Q_{A}} </math>, is then defined as the difference between the number of electrons on the isolated free atom, which is the atomic number <math> \mathbf {Z_{A}} </math>, and the gross atom population:
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| :<math> \mathbf {Q_{A}} = \mathbf {Z_{A}} - \mathbf {GAP_{A}}</math>
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| One problem with this approach is the equal division of the off-diagonal terms between the two basis functions. This leads to charge separations in molecules that are exaggerated. In a modified Mulliken population analysis,<ref>{{ cite doi | 10.1021/om950966x }}</ref> this problem can be reduced by dividing the overlap populations <math> \mathbf {P_{\mu\nu}} </math> between the corresponding orbital populations <math> \mathbf {P_{\mu\mu}} </math> and <math> \mathbf {P_{\nu\nu}} </math> in the ratio between the latter. This choice, although still arbitrary, relates the partitioning in some way to the electronegativity difference between the corresponding atoms.
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| Another problem is the Mulliken charges are explicitly sensitive to the basis set choice.<ref name = "NPA">{{cite journal | doi = 10.1063/1.449486 | journal = J. Chem. Phys. | title = Natural population analysis | author1 = A. E. Reed | author2 = R. B. Weinstock | author3 = F. Weinhold | year = 1985 | volume = 83 | issue = 2 | pages = 735–746|bibcode = 1985JChPh..83..735R }}</ref> This problem is addressed by Natural Population Analysis <ref name = "NPA" /> and other modern methods for computing net atomic charges.
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| == See also ==
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| * [[Partial charge]], for other methods used to estimate atomic charges in molecules.
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| ==References==
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| <references/>
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| [[Category:Quantum chemistry]]
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