General expression for the spatial partitioning of the moments and

J. Phys. Chem. 1993,97, 12760-12767

12760

General Expression for the Spatial Partitioning of the Moments and Multipole Moments of Molecular Charge Distributions Keith E. Laidig College of Chemistry, University of California, Berkeley, California 94720 Received: June 22. 1993e We demonstrate that the moments and multipole moments of a molecule are expressible as a sum of atomic contributions, each of which is based upon the spatial distribution of charge within the atoms. It is shown that the origin of any molecular moment and its corresponding multipole moment may be readily understood from an investigation of thesecontributions. The use of thespatially defined atoms of the theory of atoms in molecules correctly partitions any molecular moment or multipole moment into physically meaningful atomic contributions. The general expressions for the summation of atomic components to yield molecular moments and multipole moments are present and their use is demonstrated by the construction and investigation of the molecular dipole, second, quadrupole, third, and octupole moments of the series of fluorinated methanes CH&,F,, in = 0-4. The transfer of charge from CHk, to F, as inincreases dominates the origins of the molecular moments and explains the trends of change throughout the series.

Introduction The interaction between molecules is conveniently addressed via the multipole expansion of each of the individual molecular charge distributions.' The multipole moments arise as the coefficientsof the Taylor series expansion of the inverse distance, R',between the molecular origins. These terms appear, for example, in the potential generated by an individual molecular charge distribution, 4, as expressed in

The tensor T a ~ y , .is. u proportional to symmetric to interchange of any pair of suffixes, and reduces to zero on contraction (e.g. Taay..,u = 0). This has been discussed previously,' and we merely list the general expression

Tu,, .,," = (4mo)-'VaVBVy...O K '

(2)

The first four multipole moments are then defined to be2 q = c e i = the molecular charge

(3)

i

1 , = c e i r i a = the molecular dipole

(4)

i

8, = ' / 2 c e i ( 3 r i a r i-f ir,%,& = the molecular quadrupole i

(5)

the molecular octupole (6) where ei is the charge of the particle and rta represents the a component of the displacement of the particle from the origin. These have been defined in terms of the molecular moments, the corresponding expectation values of r"; they are symmetric to interchange suffixes, and they also reduce to zero on contraction.2 Abstract published in Advance ACS Absrracrs, November 1, 1993.

0022-3654/93/2097- 12760%04.00/0

The multipole moments are thus a measure of the anisotropy of the various molecular moments of the same order; e.g., the quadrupole moment is the measure of the anisotropyof the second moment, (rarp),of the charge distribution. The theory of atoms in molecules demonstrates that the moments and multipole moments of a molecule are expressible as a sum of atomic contributions, as are all molecular properties.3 The theory has been used to partition a wide range of properties such as energetics: polarizabilities,'Y6 and magnetic susceptibilities.' It also brings to light the underlying principles behind many empirical characteristics used in chemistry! eg., bond orders9 and electronegativity.10 We present an analysis of the origin of molecular moments based upon the atomic properties predicted using subsystemquantum mechanicsas developedwithin the theory of atoms in molecules. We begin by developing the general expression of the moments of the molecular charge distribution as a summation of atomic contributions. The development is then extended to the more widely used multipole moments of the charge distribution. The theoretical basis of our partitioning scheme is then reviewed, and, as an illustration, we also investigate the atomic contributions to the molecular dipole, second, quadrupole, third, and octupole moments for the series of fluorinated methanes CH&,F,, m = 0-4. The origins of a molecular multipole moment are of general interest as a systematic understanding of the origins of molecular moments will generate predictive powers which can then be extended with confidence to systems which are beyond present theoretical and experimentaltechniques. This procedure has been used to demonstrate that the form of the charge distributions within the (26 rings of C6H6 and C6F6 have the same form despite the difference in molecular quadrupole and that the origin of the positive quadrupole in C6F6is the large transfer of charge from the origin by F.IL The curious trend in the quadrupole moment through the series COz, OCS, and CSz, going from negative to positive, is similarly uncovered.12 It was demonstrated to result from the decrease in the transfer of charge from the origin as S replaced 0, despite the much larger size of S. Further interest in this research arises from a desire to produce convenient, compact, and robust representations of the charge distribution to be used to simulate interactions using dynamical simulations, etc. The multipole representation of the charge distribution has been very successful in reproducing the electrostatic properties and using them to investigate molecular interaction.'sl5 This will provide procedures by which thevarious multipole moments of a very large system could be constructed 0 1993 American Chemical Society

The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12761

Spatial Partitioning of Molecular Moments using transferable components and the molecular potentials generated.13J'

General Expression for Moments of a Molecular Charge Distribution

The general expression for an arbitrary moment, distribution of charges is

s',"B..,=

p ) ,of a

(7) When expanded to consider the nuclei and electrons within the distribution eq 7 becomes (~JBr,"*'Y)

We have defined q(52) as the atomic charge, M,(fl) is the a component of the atomic first moment, and Qap(fl) is the ab component of the atomic second moment, and so on. Each of these atomic, tensorial contributions to the molecular property tensor is based upon the form of the charge distribution within the basin of the atom and the relative location of the atom to the origin. As discussed below, any moment may be seen as the tensorial summation of these associated atomic contributions which are based upon the atomic characteristics. The first four molecular moments are explicitly expressed in terms of their contributionsin Appendix 1, and the second moment is expanded in Appendix 3 as an example of this procedure.

General Expression for Multipole Moments of the Charge where Zn is the charge on the nucleus, is the a component of the displacement of nucleus fl from the origin of the property evaluation,ra is the a component of the displacement vector from the same origin, and p(r) is the electronic charge density in electrons per unit volume. This may be expressed in terms of atomic contributions by breaking the integration over all space into integrations over the regions of space associated with each atom within the molecule, the atom's basin, by making the substitution r = P + X" to yield

@$$ (1 1) Equation 10 may be rewritten as

P(.B)$(Q)$..3

+ + P(agy...~)~~~..,(n)~ + *.*

l$7...u(Q)l (12) where we haveused the followingdefinitions for thevarious atomic moments3J1

We have used the repeated gradient notation to imply the construction of the multiple moment in direct analogy to the Tap7....tensors of eq 2. Once the particular moment has been chosen, the expansion may be completed using the procedures discussed in the previous section for the molecular moments by substitution of the various atomic moments, etc. The quadrupole and octupole moments are given in terms of the atomic contributions in Appendix 2, and the quadrupole moment is explicitly expanded as an example in Appendix 3. The reader will note that this derivation of atomic contributions has not required that the atom to be used for this partitioning arise from a particular theory or model. It requires only that each atom have a distinct, spatial volume and that the partitioning exhaust all space. This derivation, a partitioning of the tensor representingmolecular moments into a muticenteredsummation, demonstrates that a molecular moment is not merely the summationof the associated atomic moments, exceptingthe trivial case in which the atoms all have spherical distributions. The general expansion of the molecular moment demonstrates that each order of the Taylor's series expansion depends upon the displacement of the lower order terms. This derivation also demonstrates that the simple separation of charge model, often used to describe dipole and higher quadrupole moments, is only appropriate when the atoms have spherical charge distributions and thus cannot be used in a general manner. We submit that

Laidig

12762 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993

TABLE I: Atomic Contributions to the Molecular Population and Molecular Energetics of CH,, CHsF, CH2F2,CHFk and CF4 Using the HF/631G**//HF/631G** Wave Functions. N(Q) E(Q)

C 5.7542 -37.6098

Hz 1.0614 -0.6480

H3 1.0614 -0.6480

N(Q) E(Q)

C 5.1327 -37.2698

Fz 9.7428 -99.7927

H3 1.0414 -0,6591

C

F3 9.7443 -99.8775

Hz

Fa 9.7445 -99.9535

N(Q) E(Q)

4.4961 -36.8231

Fz 9.7443 -99.8775

N(Q) E(Q)

C 3.8086 -36.2697

0.9520 -0.6437

N(Q)

E(Q) 0

C 3.0522 -35.6144

Fz 9.7373 -100.0076

CH4 H4 1.0614 -0.6480 CHaF H4 1.0414 -0.6591 CHzFz H4 1.0073 -0.6609 CHF3 F4 9.7445 -99.9535

F3 9.7373 -1 00.0076

CF4 F4 9.7373 -100.0076

Hs 1.0614 -0.6480 H5 1.0414 -0.6591 H5 1.0073 -0.6609 Fs 9.7445 -99.9535 F5 9.7373 -1 00.0076

sum 9.9999 -40.2017

mol val 1o.oooo -40.2017

sum

mol Val

17.9998 -139.0398

18.0000 -1 39.0397

sum

mol Val

25.9994 -237.9000

26.0000 -237.8998

sum

mol Val

33.994 1 -336.7738

34.0000 -336.7734

sum

mol Val

42.0014 -435.6449

42.0000 -435.6452

N ( Q ) is the atomic population, and E(Q) is the atomic total energy. All values are listed in atomic units.

any spatial partitioning scheme must recover these relationships if the regions defined are to be physically meaningful.

bounded by surfaces of zero flux.

Quantum Mechanics of Open Systems

Similarly, the atomic contributions to the second moment, eq 15, is the sum of expectation values multiplied by the associated displacement vectors. The form of the spatial distribution of charge within each atom is used as the basis to describe the spatial distribution of charge within the molecule. This brings a physical and quantum mechanical basis to the investigation of the origin of the molecular moment which might be otherwise difficult to uncover. In addition to being the expectation values of physical observables, the atomic properties predicted by theory enjoy the advantage of transferability between similar systems, as do all the atomic properties predicted using the theory of atoms in m o l e c ~ l e s . ~ J ~As J ~the J ~ atoms of theory are defined in real space, they represent the physical characteristics of the charge distribution within that space. And, in as far as two atoms are the same, so too are their contributions to molecular properties. This is the physical basis for the success of the presented work and the theory of atoms in molecules in general, namely, that the formof thechargedistribution within a regionof spacedetermines the contribution of that region to the properties of the molecule. Thus to the degree that the charge distribution in two atomic basins are the same, so too will be their contributions to the physical properties of the molecule. The atoms bounded by surfaces of zero flux in the gradient vector field of the charge density have been shown to be nearly constant in their properties between different systems and that other choices of surfaces for partitioning decrease the constancy of atomic properties between system.l*J9 The transferability of moments and their polarizabilities has been demonstrated6 and is the topic of on-going research.Z0

The theory of atoms in molecules generalizes quantum mechanics to subsystems, atoms, or groupings of atoms, within a total system.3J6 Atoms are open quantum mechanical systems that are bounded by a surface through which there is no flux in the gradient vector field of the charge density at each point of the surface Vp(r).n(r) = 0 Vr E S(r) (20) This boundary condition appears as a constraint in the generalized variation of the quantum action integral and is equally valid for the total system as well as each atom within the system. Variation of the quantum action integral subject to the constraint in eq 20 yields the principle of stationary action for an open system,16 Q, 6G[\k,Q] =

[&,fi)Q+ complex conjugate}

(21)

G[\k,Q]is Schrainger's energy functional and P i s an operator which induces infinitesimalchange in \k and thus in the functional, G. Equation 21 defines the physical observables, their average values, and the theorems which determine the mechanics of a subsystem in a stationary state. The averaging of an observable over an open system is denoted by the symbol ( )Q which requires integration over all spin coordinates and integration over all spatial coordinates of theelectrons but one. This yields a property density at each point in space which is the average of the property over the motions ofall the particles within thesystem. When integrated over the basin of the atom, this yields the atomic average value of the property. In this manner, the properties and mechanics of the atoms, and indeed the total system, are predicted by quantum mechanics.3J6 We have chosen to use the theory of atoms in molecules to provide our spatial partitioning because of the quantum mechanical basis of the atoms so defined. It has been demonstrated that the populations defined by the theory of atoms in molecules are physical 0bservab1es.I~ This proof may be generalized to show that any moment of the quantum atom is also a physical observable. The atomic moments defined in eqs, 13-16 will be determined by integration of the appropriate property density over the atomic basins bounded by zero-flux surfaces as defined in eq 20. For example, the atomic second moment is the expectationvalue of the operator ruro,which is the integral of the density weighted by the operator evaluated over the atomic basin

Methodology and Results Each molecule was optimized using the 6-3 1G**basis21,22 within the CADPAC5 program package23 constrained to the following symmetries: methane and tetrafluoromethane as Td, methyl fluoride and trifluoromethane as C,,, and difluoromethane as Cb,The wave functions were produced a t the same level of theory using the ANVIL5 properties package24 and then analyzed using the AIMPAC program ~uite.2~ The atomic contributions to the molecular energetics for all the molecules are listed in Table I, and their summation are accurate throughout the series to 0.006 electrons in population and within 0.3 kcal mol-' in total energies. The atomic properties were determined by integration

Spatial Partitioning of Molecular Moments


TABLE 11: Atomic Contributions to the Nonzero Molecular Dipole, Second, Quadrupole, Third, and Octupole Momentcl of Methaneb Y

properties

(W

mol Val

C

H2

H3

H4

HS

sum

-2.473 0.138 0.073 0.065 -0.824 -0.086 -0.077 -0.584

-2.473 0.138 0.073 0.065 -0.824 -0.086 -0.077 -0.584

-2.473 -0.138 -0.073 -0.065

-2.473 -0.138 -0,073 -0,065

-18.645 O.Oo0 O.Oo0 0.000

-18.644 O.OO0

-0.824 -0.086 -0.077 -0.584

-0.824 -0.086 -0.077 -0.584

-6.215 -0.344 -0,308 -5.254

-6.215

QAn)

-8.752 O.Oo0 0.000 0.000 -2.9 18 0.000 0.000 -2.918

e 2 :

-0.001

0.000

0.000

O.Oo0

0.000

-0.001

O.OO0

26X3$X

0.98 1 0.000 0.000

-0.013 -0.102 -0.091 0.097 0.07 1

-0.013 -0.102 -0.091 0.097 0.071

-0,013 -0.102 -0.09 1 0.097 0.071

0.930 -0.407 -0.365 0.390 1.263

0.914

0.98 1

-0.013 -0.102 -0.09 1 0.097 0.071

2.500

-0.054

-0.054

-0.054

-0.054

2.285

2.285

4

dQ)X

MQ) Q:' 4(Q)X1 XMAQ)(2)

g2;;;;3) Qxyz 0

0.000

All values are listed in atomic units. The properties are labeled in accord with the depicted coordinate system. Values in parentheses are the number of particular terms found in the molecular expansion.

TABLE III: Atomic Contributions to the Nonzero Molecular Dipole, Second, Quadrupole, Third, and Octupole Moments of Fluoromethaneb V

properties

(W 0

%OX MAQ) @X

me *MAW (2) Q"z

QAQ)

en 8"

822

R& R;Yz

:(;)e

xQ"M,(Q) (3) $Q,SQ) RAQ)

no ,8""

Qf2

SIPY

QE

(3)

mol Val

C

F2

H3

H4

H5

sum

-6.742 0.483 1.114 -0.63 1 -2.358

-1 1.941 0.642 0.961 -0.320 -3.885

-2.510 -0,115 -0.081 -0.034 -0.721

-2.510 -0.115 -0,081 -0.034 -0.721

-2.510 -0.115 -0.08 1 -0.034 -0.991

-26.214 0.780 1.832 -1.053 -8.675

-2.027 1.431 -0.810 -1.838 -0.166 0.331

-4.171 -1.244 0.4 14 -3.754 0.143 -0.286

-0.889 -0.159 -0.067 -0.597 0.175 -0,078

-0.889 -0.159 -0.067 -0,597 0.174 -0.078

-0.889 -0.159 -0.067 -0.597 -0.232 -0.078

-8.864 -0.289 -0.596 -7.384 0.094 -0.189

-8.865

0.71 1

0.042

2.176

2.176

-4.426

0.679

0.679

-26.216 0.779

-8.676

0.095 -0.189

-3.180

4.959

-1.413

-1.413

-1.802

-2.849

-2.849

-0.709

-0.042 4.959

0.758

0.758

-1.673

-1.673

14.388 1.611 -0.536 4.859 -0.192 0.105 0.245 -0,105 0.245 -0.490

-4.430 -0,310 -0.130 -1.168 -0.225 0.4 19 0.226 0.222 -0.424 0.198

-4.430 -0.310 -0.130 -1.168 -0.225 -0.71 1 0.226 -0.155 -0.424 0.198

-1.442 -1.283 -4.430 -0.310

-0.677 -2.847 -9.007 2.518 -1.961 -1.007 -2.606 1.695 0.230 -1.694 0.234 -0,464

-0.679 -2.849 -9.009

-3.179 -1 0.106 1.838 -1.041 -2.361 -1.740 1.775 0.281 -1.774 0.286 -0.567

-0.130

-1.168 -0.225 0.108 -0.749 0.118 0.551 0.198

1.698 0.214 -1.698 0.214 -0.463

All values are listed in atomic units. The properties are labeled in accord with the depicted coordinate system. Values in parentheses are the number of particular terms found in the molecular expansion. a

Laidig


TABLE Iv: Atomic Contributions to the Nonzero Molecular Dipole, Second, Quadrupole, Third, rad Octupole Moments of DifluoromethanPs

t'

properties

(W ll

k

dQ)X M Q )

Qx: Qh QfZ

mX2 X M z ( Q ) (2) QZ*(Q)

3e R:XZ

C

Fz

Fs

H4

H5

-4.057 -1.279 -0.476 -0.802 -1.861 -1.251 -0.945 0.151 0.254 -1.064 -0.763 0.152 0.611 0.603

-15.009 1.110 1.335 -0.226 -3.849

-15.009 1.110 1.335 -0.226 -3.849 -5.749

-1.970 -0.067 -0.006 -0,061 -0.775 -0.573

-1.970 -0.067 -0.006 -0.06 1 -0.775 -0,573

-5.41 1 -2.395 0.405 -3.825 1.731 -1.118 -0.613 6.882

-0.622 -0.005 -0.050 -0.516 -0.177 0.125 0.052 -0.528

-0.622 -0,005 -0,050 -0.516 -0.177 0.125 0.052 -0.528

-0.106 22.468 4.296 -0.726 6.862 -0.237 -2.022 3.537 -1.515

-0.608 -1.693 -0.004 -0.041 -0.423 -0.298 0.094 -0.105 0.011

-0,608 -1.693 -0.004 -0.041 -0.423 -0.298 0.094 -0.105 0.011

-0.382 -0.380 -0.048 -0.08 1 0.508 -1.615 1.588 -0.876 -0.712

-5.749 -5.41 1 -2.395 0.405 -3.825 1.731 -1.118 -0,613 6.882 -0.106 22.468 4.296 -0.726 6.862 -0,237 -2.022 3.537 -1.515

sum -38.013 0.806 2.182 -1.376 -11.107

mol Val -38.0 19 0.807

-11.109

-13.895 -13.011 -4.649 0.963 -10.288 2.346 -1.836 -0.510

-13.896 -13.014

13.312 16.614 41.169 8.536 -1.614 13.386 -2.685 -2.268 5.988 -3.720

13.312 16.613 41.175

2.346 -1.836 -0.511

-2.269 5.983 -3.714

a All values are listed in atomic units. The properties are labeled in accord with the depicted coordinate system. b Values in parentheses are the number of particular terms found in the molecha; expansion. of the appropriately weighted density over the atomic basins bounded by zero-flux surfaces. Only the first nonzero multipole moment is origin independent, but for convenience the center of mass is chosen as the origin of the property evaluation. In this I J c * F work it is our intentionto focusupon the propertiesof the molecular moments predicted using the HF/6-3 lG**//HF/6-3 1G**wave function rather than try to reproduce the experimentallyobserved moments. Theatomiccoordinates andpropertiesof themolecules, mlecular dipole " e n t as well as atomic contributions to the molecular moments, are q(FJ'X.fP1 listed in Tables 11-VI. The molecular properties are recovered MJHI 0 (x3) by the summation of atomic contributions up to 0.005 au in Atomc first moment general, but to within 0.02 au in the worst cases, Qxxx and Qxyy contributionsto the MJCJ -< mlrmlar dipole mment for CHF3. M,fFJ a

I i

-Y

Discussion In order to understand the contributions toward the molecular moments of the molecules in the series, we shall first investigate the physical meaning of this tensorial summation of contributing atomic moments. For example, we consider the construction of the molecular dipole moment of methyl fluoride from the appropriate atomic contributions.26 The z-component of the molecular dipole moment of methyl fluoride is constructed using the following terms

The appropriate contributions and their summation are listed in Table 111, and the tensorial summation of the individual atomic components is graphically depicted in Figure 1. There is a large transfer of charge to F from CHI which generates large chargetransfer contributions from the C and F atoms. Thedisplacement

Molecul~dipole moment

t PZ

Figure 1. Graphical depiction of the vectorial summation of the atomic contributions which lead to the molecular dipole moment in CH3F. The

charge-transfer contributions are represented by solid arrows, the first moment contributions by hollow arrows, and the magnitude of the contribution represented by the length of the arrow. of the F atom's net charge of -0.743 e, a distance of -1.294 au from the origin, generates a charge-transfer contribution to the pzvector of +0.961 au. Similarly, the net charge on C of +0.847 e is displaced +1.285 au and thus generates a further +1.114 au contribution to the same component. The three H atoms have a much smaller net charge, and despite their large relative displacement from the origin, they each make only a small contribution in opposition to the of C and F. The sum of chargetransfer contributions to the molecular moment is +1.832 au. The polarizationof the Fatomaway from theCH3groupproduces an opposing contribution to the charge transfer of -0.63 1 au. The



TABLE V Atomic Contributions to the Nonzero Molecular Dipole, Second, Quadrupole, Third, and Octupole Moments of Trifluoromethad

t'

C

H2

Fs

F5

sum

mol val

-1.626

-13.715

-13.717

-13.717

-46.074

-46.094

0.008 0.116 -0.108

0.206 0.365 -0.160

0.206 0.366 -0.160

0.206 0.365 -0.160

0.670 2.045 -1.376

0.673

MZ(W

-3.301 0.045 0.833 -0,789

Q",

-0,842

-0.521

-4.372

-4.372

-6.065

-16.172

-16.182

-0.583 0.280 -0.261 -0.342

-3.843 -0.179 0.078 -3.821

-3.843 -0.179 0.078 -3.821

-3.843 -0.179 0.078 -3.821

-13.727 0.059 -0.326 -13.135

-13.730

Qzz(W

-1.614 0.317 -0.300 -1.331

ezz

0.387 -0.771

0.031 -0.062

0.300 1.093

0.300 1.094

-2.239 1.093

-1.221 2.447

-1.226 2.452

0.138

0.027

13.619

13.620

-29.390

-1.985

-1.972

-0.914

-1.447

1.808

1.808

1.990

3.245

3.247

-0.142

-0.027

5.496

5.497

-8.844

1.983

1.972

-0.915

-1.447

1.930

1.930

1.747

3.245

3.247

-2.684 0.120 -0.114 -0.506 -0.945

-4.078 0.676 -0.630 -0.825 -0.385

5.470 0.088 -0.038 1.875 -0.127

5.470 0.088 -0,038 1.875 -0.127

5.470 0.088 -0,038 1.875 -0,127

9.654 1.060 -0.859 4.294 -1.711

9.662

0.351 -0.029 -0.353 -0,030 0.059

0.068 -0.135 -0.068 -0.135 0.269

-0.730 -0.084 2.146 0.220 -0.136

-1.344 -0,084 1.945 0.221 -0.137

-3.305 0.372 1.288 -0.237 -0.136

-4.961 0.040 4.959 0.040 -0,080

-4.930 0.040 4.930 0.040 -0.079

properties

(W

4 dQ)X

z@

me2

X M Z ( Q )(2)

R:XZ

RY :Y

zR ; e z z

dWXJ

flww (3) $'Qzz(W

(3)

Rzzz(Q)

fla QfX

$= nvy

E

F4


TABLE VI: Atomic Contributions to the Nonzero Molecular Dipole, Second, Quadrupole, Third, and Octupole Moments of Tetrafluoromethane4b

t'

properties

C

F3

F4

Fs

sum

mol val

-13.596

-13.596

-13.596

-13.596

-55.630

-55.629

0.791 1.047 -0.256

0.791 1.047 -0.256

-0.791 -1.047 0.256

-0.791 -1.047 0.256

O.Oo0 O.Oo0 0.000

O.OO0

-4.532 -1.487 0.364 -3.772 O.Oo0

-4.532 -1.487 0.364 -3.772 0.000

-4.532 -1.487 0.364 -3.772 0.000

-4.532 -1.487 0.364 -3.772

-18.540

O.Oo0

-18.540 -5.949 1.454 -15.498 0.006

-0.483 -2.112 0.516 0.003 0.073

-0.483 -2.112 0.516 0.003 0.073 -1.207

-0.483 -2.112 0.516 0.003 0.073 -1.207

-0.483 -2.112 0.516 0.003 0.073 -1.207

-1.811 -8.449 2.065 0.01 1 0.410 -4.527

-1.813

F2

-1.207

0.OOO

-4.533



polarization of the C atom toward F adds further to this opposition as do the small polarizations of the H atoms toward the C atom, yieldinga total polarizationcontributionof-1.053 au. Thevector sum of these two types of terms yields the z-component of the molecular dipole moment of CH3F of +0.780 au. The excellent agreement with the molecular value of +0.779 au demonstrates clearly that the molecular moment is the sum of the atomic contributions listed in eq 23. In a completely analogous manner any of the moments and multipole moments of the charge distributions may be constructed from atomiccontributions. Each atomic contribution arises from the physical properties of the individual atoms, the atomic moments, and the displacement of the nuclei from the origin of property evaluation. This partitioning brings to light the physics underlying the origin of the molecular moment or multiple moment -27 The trends in the moments through the series CH&,F,, m = 0-4,are related to the number of F atoms introduced into the molecule. Methane and methyl fluoride have large contributions to their molecular moments from the atomic moments of the C atom. As the number of F atoms increase, so does the transfer of charge away from C and thus the decrease of contributions from the C atomic moments. The increase in the number of F atoms leads to a greater and greater importance in the chargetransfer terms associated with the F atoms as well as the second moments of the F atoms. The dipole moment is influenced by the increased importance of the charge-transfer terms, q(Q)e, and this quickly becomes the dominant feature throughout the series. For the second and quadrupole moments, it is a combination of the dominance of charge transfer and the second moments of the F atoms that is the crucial factor in determining the molecular moments. The third and octupole moments are similarly dominated by the charge-transfer and displaced second moment contributions as the number of F atoms increased across the series. These trends are readily understood in terms of the physical characteristics of the atomic moments. The large transfer of charge from CH, to F, emphasizes the contribution of these terms to the molecular moments. The resulting atomic distributions in C and H polarized away from the F, creating opposing contributions, but the polarization is diminished by thedecreasing amount of remaining charge within the C and H atom's basins to polarize. The F atoms do not polarize appreciable, and thus these terms are also small. The large second moments of the atomic distributions of F in comparison to H or make their contributions to the molecular moments stand out and dominate with increasing number of F. As the F atoms are displaced from the origin, their contribution to the molecular second and third moments is emphasized and play a crucial role. Conclusions This work demonstrates that the spatial partitioning the molecular moments into atomic contributions results in an insightful interpretation of the origin of these moments in terms of the atomic properties. The success of this partitioning of molecular properties is based upon the physical significance of theatoms defined by subsystemquantum mechanics. Partitioning the molecule into spatial regions and then constructing atomic contributions based upon the form of the distribution within each atomic basin bring to the fore the physical form of the molecular distribution. The derivation presented did not require a particular definition for the atoms to be used. In fact, we submit that any spatial definition of an atom must generate the molecular moments in the manner described above if the atoms are to have physical meaning. We have demonstrated that the theory of atoms in molecules successfully fulfills these requirements, generating a partitioning of the molecular moments and multiple moments into atomic contributions based upon the spatial distribution of charge within the atoms.

Laidig This derivation, a partitioning of the tensor representing molecular moments into a multicentered summation, demonstrates that a molecular moment is not merely the summation of the associated atomic moments. This derivation also demonstrates that the separation of charge model is only appropriate when the atoms have spherical charge distributions and thus should not be used in a general manner. Acknowledgment. This work was undertaken during a research associateship a t the University of Cambridge, and the author is indebted to Dr. R. D. Amos for financial support. The author is pleased to thank Dr. J. S. Andrews, Ms. L. M.Cameron, and Dr. D. B. Whitehouse for carefully reading the manuscript and contributing constructive comments. Support a t Berkeley was provided by NIH Grant No. S10 RR0561-01 for computational facilities and by a postdoctoral fellowship with Prof. A. Streitwieser. Appendix 1. The Zero, First, Second, and Third Moments of a Molecular Charge Distribution The atomic expansion of the zero, first, second, and third moments of a molecular charge distribution are listed, using the general expression in eq 12 and n = 0, 1,2, and 3, respectively. The definitions in eqs 13-16 have been used. 4 = &(Q) n

(Al.l

Appendix 2. The Quadrupole and Octupole Moments of a Molecular Charge Distribution The atomic expansions of the quadrupole and octupole moments of a molecular charge distribution are listed using eq 19 with n = 1, 2, 3. The definitions in eqs 13-16 have been used. The atomic expansions of the monopole and dipole moments are the same as those given in Appendix 1 for the zero and first moments.



Appendix 3. Explicit Expansion of the Second and Quadrupole Moment We consider the construction of the CUBcomponent of the molecular second moment as an example:

Qt, = [q(Q)p& + M,(Q)X; + M,(Q>X + Q,&Q)l (A3.1) The atomic contribution to the second moment is built from which are the displacements of charge-transer terms, q(Q)p&, the atomic charge from the molecular origin, the two displaced first moment terms, M,(Q)$ and M , ( Q ) g , which are the displacement of the atomic first moments from the molecular origin, and the atomic second moments, Qa,(Q). The molecular second moment tensor is then constructed by summing these terms over the atoms for each component e., Similarly, the CUBcomponent of the quadrupole moment is expanded in the following manner from eq 18

-

1

Equation A3.4 is the general expressionof the quadrupole moment which when expanded into atomic based integration and expansion of the grad operators yields

3(r,"+

J"(' + P)5

X>

General expression for the spatial partitioning of the moments and

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