Matta, Boyd. The quantum theory of atoms in molecules

82 3 Atomic Response Properties

Fig. 3.14 E ect of finite (x @0:2 a.u.) asymmetric stretching vibration on CO2 in terms of electron density contours, interatomic surfaces (bold), and bond paths (semi-bold). Dotted lines correspond to asymmetrically stretched geometry whereas solid lines correspond to equilibrium geometry.

3.8

Atomic Nuclear Virial Energies

The electronic energy of an atom in a molecule, EeðWÞ, equivalent to minus the electronic kinetic energy of the atom, TðWÞ, has a number of important characteristics [1], including:

Well defined. The Hamiltonian (K) and Lagrangian (G) kinetic energy densities integrate to the same atomic electronic kinetic energy.

Total energy additivity at stationary point geometries. The sum of the EeðWÞ in a molecule at a stationary point geometry is equal to the total energy E of the molecule because of the molecular virial theorem.

Transferability. The atomic electronic energies for di erent atoms – and their corresponding kinetic energy and virial field distributions – are similar in a manner which parallels the atomic charge distribution and which is consistent with experimentally based atomic and group additivity schemes.

3.8 Atomic Nuclear Virial Energies 83

An atomic electronic virial theorem is satisfied:

EeðWÞ ¼ TðWÞ ¼ ð1=2ÞVeðWÞ, where VeðWÞ is the atomic basin average of the virial of the electronic Ehrenfest force density plus the atomic surface average of the virial of the electronic pressure density, i.e. the atomic electronic potential energy contribution.

Interpretable in terms of local, real space energy densities.

An open ‘‘problem’’ in QTAIM, however, is that for molecules at non-stationary point geometries the sum of the atomic electronic energies EeðWÞ does not equal the total molecular energy E [1] because the forces on the nuclei (and hence the nuclear virial contribution to the energy) are not zero, i.e. E 0 Ee. This is easily seen from the molecular virial theorem [16], i.e., the hypervirial theorem for the

full (electronic þ nuclear) virial operator ^ :

V

				Ne
^ ^	^			Xi
V ¼ Ve þ Vn		¼ ð1=2Þ ½p^i ri þ ri p^i &
				¼
				1
	Na
	X			^
þ ð1=2Þ			^		ð38Þ
		¼	½PA RA þ RA PA&
	A		1

^	^	^
ði=hÞhcj½H; V&jci ¼ ði=hÞhcj½E; V&jci
	^	^	^
	¼ 2hcjTjci þ hcjVnejci þ hcjVee

j i þ h j^ j i c c Vnn c

Na

X

¼	¼	RA ‘AE
A		1
¼ 2T þ Vne þ Vee þ Vnn ¼ 2T þ V ¼ T þ E ¼ W			ð39Þ

Rearranging Eq. (39) gives:

E ¼ T þ W ¼ ð1=2ÞðV þ WÞ ¼ Ee þ W

ð40Þ

The energy W may be called the ‘‘nuclear virial’’ contribution to the total molecular energy:

Na			Na
X			X			ð41Þ
W ¼ RA ‘AE ¼				¼	RA GA
A	¼	1	A		1

where GA is the familiar energy-gradient based force on the nucleus A (which is, unfortunately, not equal to the Hellman–Feynman electrostatic FA force on nucleus A in typical ab initio calculations since they typically violate the HellmanFeynman electrostatic theorem):

84 3 Atomic Response Properties

GA ¼ ‘AE

ð42Þ

In terms of atomic electronic energies EeðWÞ ¼ TðWÞ we have:

Na		Na
X1		X1			ð43Þ
E ¼ T þ W ¼	¼	TðWÞ þ W ¼	¼	EeðWÞ þ W
W		W

For molecular geometries at which GA ¼ 0 for all nuclei (e.g. equilibrium and transition state geometries), W is zero and atomic electronic energies EeðWÞ are additive to give the total molecular energy E ¼ Ee.

For non-stationary point geometries this is not true and it is therefore important to be able to define a physically reasonable atomic contribution, WðWÞ, to W so that the total energy additivity of atomic energies is preserved as a molecule vibrates, reacts, undergoes a vertical electronic transition or otherwise deviates from a stationary point geometry. In addition, even at equilibrium geometries a definition for WðWÞ is necessary if one wishes to calculate atomic contributions to molecule response properties defined in terms of energy derivatives with respect to nuclear coordinates (e.g. vibrational frequencies).

	Na
W ¼	X1		ð44Þ
		WðWÞ
	W	¼
EðWÞ ¼ EeðWÞ þ WðWÞ ¼ TðWÞ þ WðWÞ			ð45Þ
	Na
X1			ð46Þ
E ¼		EðWÞ

W¼

In the limit of a stationary point geometry, where W ¼ 0, each WðWÞ should separately equal zero so that the total atomic energy EðWÞ reduces to the atomic electronic energy EeðWÞ, which is already known to behave correctly at stationary point geometries:

limW!0½WðWÞ& ¼ 0

ð47Þ

An obvious definition for WðWÞ which satisfies Eq. (47) would be:

WI ðWÞ ¼ RW GW

ð48Þ

where GW means the energy-gradient-based force on the nucleus of atom W and the subscript ‘‘I’’ is used to distinguish this definition from an alternative definition considered later.

As indicated by the underline, the definition for WIðWÞ in Eq. (48) is origindependent and is, therefore, not directly useful. It also has the physically unrea-

3.8 Atomic Nuclear Virial Energies 85

sonable characteristic that if the force on the nucleus of an atom is zero, its nuclear virial energy contribution is necessarily zero. To see that this is unphysical it is important to realize that the total force on a nucleus in a molecule is a molecular property, and therefore one to which each atom contributes [1]. This is evident from the Hellman–Feynman electrostatic theorem [16], discussed later in this section. Equation (48) does, however, provide a good starting point for defining a more physically reasonable WIðWÞ. Because of translation invariance [16], the sum G of the energy-gradient-based forces GW is zero, i.e. G is a ‘‘null’’ molecular property:

Na
X1		ð49Þ
G ¼	GW ¼ 0

W¼

Thus, the method of partitioning apparently origin-dependent atomic properties outlined in Sections 3.2 and 3.3 can be used to define an origin-independent WIðWÞ based on the WIðWÞ given in Eq. (48):

	NbðWÞ
GW ¼	X1		ðWjLÞ	ð50Þ
		GW
	L	¼
		NbðWÞ
		X1	½RW RbðWjLÞ& GWðWjLÞ	ð51Þ
WIðWÞ ¼

L¼

The force vector GWðWjLÞ may be interpreted as a ‘‘bond force’’, or a bond contribution to the energy-gradient-based force on the nucleus of atom W. With this definition of an atomic nuclear virial energy, the corresponding definition of a total atomic energy EIðWÞ in a molecule at an arbitrary geometry becomes:

NbðWÞ
X1
EIðWÞ ¼ TðWÞ þ WIðWÞ ¼ TðWÞ þ ½RW RbðWjLÞ& GWðWjLÞ ð52Þ
L	¼

All of the terms in Eq. (52) are readily calculated from standard ab initio wavefunctions. For the additivity of Eq. (46) to be obtained in practice, however, the molecular virial theorem, Eq. (39), must be satisfied. Of course, satisfaction of the molecular virial theorem is also necessary at stationary point geometries for atomic energy additivity to be obtained in practice.

Unfortunately, the molecular virial theorem is not satisfied by wavefunctions from typical ab initio calculations [16]. It is, however, relatively straightforward to variationally improve any ab initio calculation to satisfy the molecular virial theorem using what this author calls ‘‘self-consistent virial scaling’’ (SCVS), which involves variationally scaling the electronic and nuclear coordinates of a wavefunction to satisfy the molecular virial theorem in a manner which is ‘‘selfconsistent’’ with the ab initio method of choice. At the Hartree–Fock SCF level,

86 3 Atomic Response Properties

SCVS simply means optimizing the coordinate scaling parameter simultaneously with the MO coe cients – and geometry, if desired. The net result of SCVS is a legitimate ab initio wavefunction or first-order density matrix and molecular geometry that satisfies the molecular virial theorem.

Further information about the contributions to WIðWÞ can be obtained by using the Hellman–Feynman electrostatic theorem [16], which is just the hypervirial

^

theorem for nuclear momentum operator PA:

^ ^	^		^
ði=hÞhcj½H; PA&jci ¼ ði=hÞhcj½E; PA&jci ¼ hcjð‘AVneÞjci þ ‘AVnn
¼ ZA"ð drrðrÞðr RAÞjr RAj 3 þ		Na	ZBðRB RAÞjRB RAj 3#
		X
		B0A
¼ FA ¼ ‘AE ¼ GA			ð53Þ

In words, the electrostatic force FA on nucleus A is equal to the energy gradient based force GA when the Hellman–Feynman electrostatic theorem is satisfied. Thus, if the Hellman–Feynman electrostatic theorem is satisfied, WIðWÞ can be expressed as:

NbðWÞ
X1	ð54Þ
WIðWÞ ¼ ½RW RbðWjLÞ& FWðWjLÞ

L¼

where FWðWjLÞ is the contribution from the bond between atoms W and L to the electrostatic force on the nucleus of atom W.

In practice, unfortunately, the Hellman–Feynman electrostatic theorem is not satisfied in typical ab initio calculations [16], and improving typical ab initio calculations to produce legitimate wavefunctions which satisfy the Hellman– Feynman electrostatic theorem is somewhat more di cult than for the molecular virial theorem.

Practical di culties aside, the Hellman–Feynman electrostatic theorem does provide an alternative way to define a WðWÞ which should be considered. Combining Eqs (41) and (53) the total nuclear virial energy W can be expressed as:

Na

Na

X

X

RA FA

W ¼

¼

RA GA ¼

¼

A

1

A

1

Na

ZARA

"

drr r

Þð

r

RA

Þj

r

RA

j

3

¼

ð

ð

X

A

¼

1

þ

Na

ZBðRB RAÞjRB RAj 3#

ð55Þ

X

B0A

3.8 Atomic Nuclear Virial Energies 87

The contribution, FAðWÞ, of atom W to the electrostatic force FA on nucleus A is easily determined (without the Hellman–Feynman electrostatic theorem, the contribution of W to GA is not easily determined):

ð

FAðWÞ ¼ drrðrÞðr RAÞjr RAj 3 þ ZAZWðRW RAÞjRW RAj 3 ð56Þ

W

Thus, W can be expressed as:

Na		Na	Na
X		X X			ð57Þ
W ¼	RA FA ¼	¼	¼	RA FAðWÞ
¼	1			1
A		A	1 W

Using this expression, an origin-dependent atomic contribution, WIIðWÞ, to W can be defined:

	Na			Na			Na
	X			X			X
	WIIðWÞ ¼	¼	RA FAðWÞ ¼		¼	ðRA RWÞ FAðWÞ þ RW		¼	FAðWÞ
	A		1	A		1	A		1
	Na
	X								ð58Þ
	¼ ðRA RWÞ FAðWÞ þ RW FTotðWÞ
	A	¼	1

where FTotðWÞ is defined as the sum of the electrostatic forces that the charge distribution (electron and nuclear) of atom W exerts on all the nuclei. When the Hellman–Feynman electrostatic theorem is satisfied, the total FTot is a ‘‘null’’ molecular property, because of translational invariance [16]:

Na
X1		ð59Þ
FTot ¼	FTotðWÞ ¼ 0

W¼

An origin-independent WIIðWÞ can therefore be expressed according to Sections 3.2 and 3.3 as:

Na			NbðWÞ
X			X
WIIðWÞ ¼ ðRA RWÞ FAðWÞ þ				¼	½RW RbðWjLÞ& FTotðWjLÞ ð60Þ
A	¼	1	L		1

and the corresponding total atomic energy EIIðWÞ is given by:

EIIðWÞ ¼ TðWÞ þ WIIðWÞ

ð61Þ

To summarize the meaning of the terms in Eq. (60), FAðWÞ is the electrostatic force that the charge distribution (electronic and nuclear) of atom W exerts on the

88 3 Atomic Response Properties

nucleus A while FTotðWjLÞ is the contribution from the (directed) bond between atoms W and L to the electrostatic force that the charge distribution of atom W exerts on the all the nuclei in the molecule.

The two definitions for WðWÞ considered here, WIðWÞ in Eq. (51) and WIIðWÞ in Eq. (60), both employ the strategy of Sections 3.2 and 3.3 to avoid apparent origin dependence, but WIðWÞ and WIIðWÞ are otherwise di erent. WIðWÞ has the advantage that it is relatively easily and reliably calculated as long as the molecular virial theorem is satisfied whereas WIIðWÞ requires satisfaction of both the molecular virial theorem and the more di cult Hellman–Feynman theorem for its application. WIIðWÞ, however, has the important advantage that its definition is more consistent with the definition of other atomic properties, and provides a means for a richer interpretation directly in terms of the charge distribution. Both definitions, and perhaps others not considered here, should be investigated.

3.9

Atomic Contributions to Induced Electronic Magnetic Dipole Moments

The electronic magnetic dipole moment, m, of a closed-shell molecule in a uniform magnetic field B is given by [13, 17]:

ð

m ¼ ð1=2cÞ ½r R0& JðrÞ dr ð62Þ

where JðrÞ is the electronic current density at a point r in real space and R0 is the arbitrary origin of the molecular coordinate system. Because the net current hJi is zero for the molecule, m is independent of R0.

Like the electric dipole moment m, the magnetic dipole moment m can be expressed in terms of atomic magnetic polarization contributions, mpðWÞ, and origin-dependent atomic ‘‘net current’’ contributions, mcðWÞ, as follows:

m ¼ ð1=2cÞ			Na		ðW½r R0& JðrÞ dr
			X1
			W	¼
¼ ð1=2cÞ			Na		ðW½r RW& JðrÞ dr þ ðRW R0Þ ðW JðrÞ dr
			X1
			W	¼
¼ ð1=2cÞ			Na		ðW½r RW& JðrÞ dr þ ½RW R0& JðWÞ
			X1
			W	¼
Na					Na
X1					X1			ð63Þ
¼	¼	fmpðWÞ þ mcðWÞg ¼				¼	mðWÞ ¼ mp þ mc
W					W

3.9 Atomic Contributions to Induced Electronic Magnetic Dipole Moments 89

where JðWÞ is the net current of atom W. As in earlier sections, underlines are used to emphasize that mcðAÞ, and hence mðWÞ, is origin-dependent. Using Eq. (7), mðWÞ can be expressed in terms of ‘‘bond current’’ contributions JðWjLÞ as:

	NbðWÞ
mcðWÞ ¼ ð1=2cÞ½RW R0& JðWÞ ¼ ð1=2cÞ½RW R0&	X1
	JðWjLÞ ð64Þ
	L¼

Using Eq. (8), a corresponding origin-independent expression, mcðWÞ, can be defined:

NbðWÞ
X1	ð65Þ
mcðWÞ ¼ ð1=2cÞ ½RW RbðWjLÞ& JðWjLÞ

L¼

where JðWjLÞ is the contribution to the net current of atom from the bond from atom W to atom L.

Note that for each term RbðWjLÞ JðWjLÞ for atom W, there is a corresponding term for atom L which cancels it, because of Eq. (8), and therefore:

Na				Na		NbðWÞ
X1				X1		X1
	mcðWÞ ¼ ð1=2cÞ				¼		½RW RbðWjBÞ& JðWjBÞ
W	¼			W		L	¼
				Na			Na
				X1			X1			ð66Þ
			¼ ð1=2cÞ		¼	RW JðWÞ ¼		¼	mcðWÞ
				W			W
	Na		Na
	X1		X1							ð67Þ
m ¼		¼	mðWÞ ¼ fmpðWÞ þ mcðWÞg ¼ mp þ mc
	W		W	¼

The meaning of mpðWÞ and mcðWÞ is essentially that the former describes the magnetic moment arising from current flowing within the atomic basin whereas the latter describes the magnetic moment from current flowing between atomic basins. When a current distribution is highly localized, mpðWÞ is larger than mcðWÞ. When a current distribution is highly delocalized, the opposite is true.

Note that all of the quantities in the expressions for mpðWÞ and mcðWÞ are uniquely determined by the molecular charge distribution and current density distributions.

In the absence of an external magnetic field B, the current density JðrÞ vanishes for closed-shell molecules [13]. For open-shell molecules, JðrÞ does not necessarily vanish even when B ¼ 0 and in such circumstances the corresponding permanent molecular electronic magnetic moment can be expressed in terms of the atomic contributions mpðWÞ and mcðWÞ described here.

903 Atomic Response Properties

3.10

Atomic Contributions to Magnetizabilities of Closed-Shell Molecules

Analogous to the electric polarizability tensor a, the magnetizability tensor of a molecule, w, is the gradient of the molecular electronic magnetic dipole moment, m, with respect to a uniform magnetic field, B, in the limit of zero field strength [13]:

w ¼ ½‘Bm&B¼0 ¼ ½iðqm=qBxÞ þ jðqm=qByÞ þ kðqm=qBzÞ&B¼0 ¼ mB

ð68Þ

where the notation tB is used to indicate the gradient of the term t with respect to B, evaluated at B ¼ 0.

From Eq. (62), w is given by:

ð

w ¼ ð1=2cÞ ðr R0Þ JBðrÞ dr ð69Þ

where JBðrÞ is the gradient of the electronic current density JðrÞ with respect to B, in the limit of zero field strength:

JBðrÞ ¼ ½‘BJðrÞ&B¼0 ¼ ½iðqJðrÞ=qBxÞ þ jðqJðrÞ=qByÞ þ kðqJðrÞ=qBzÞ&B¼0	ð70Þ
The first-order current density Jð1ÞðrÞ induced by B is:
Jð1ÞðrÞ ¼ JBðrÞ B	ð71Þ

Methods for calculating relatively accurate Jð1Þ distributions and their dependent properties, for example magnetizability tensors discussed in this section and NMR shielding tensors [18], were developed by Keith and Bader [5, 19]. Thorough and correct displays and analyses of Jð1Þ distributions were presented by Keith and Bader [20] based on methods developed elsewhere [5, 19], along with pioneering work by Gomes [21].

In terms of atomic contributions to w, from Eq. (67) we have:

Na		Na
X1		X1
w ¼	¼	mBðWÞ ¼ fmpBðWÞ þ mcBðWÞg
W		W	¼
Na			Na
X1			X1			ð72Þ
¼	¼	wpðWÞ þ wcðWÞ ¼		¼	wðWÞ
W			W

where wpðWÞ and wcðWÞ are given in terms of the current density by:

3.10	Atomic Contributions to Magnetizabilities of Closed-Shell Molecules		91
wpðWÞ ¼ ð1=2cÞ ðWðr RWÞ JBðrÞ dr		ð73Þ
NbðWÞ
X1	f½RW RbðW; LÞ& JBðrÞðWjLÞ	ð74Þ
wcðWÞ ¼ ð1=2cÞ

L¼

Note that, unlike the atomic electric polarizability tensors aðWÞ, there are no surface derivative contributions to wðWÞ, because ½‘BrðrÞ&B¼0 ¼ 0, i.e. the firstorder correction to the electron density from a magnetic field perturbation vanishes because the magnetic field perturbation term in the Hamiltonian (and hence the first-order perturbed wavefunction) are purely imaginary functions [13].

Using these formulas, Bader and Keith [6, 7] studied the atomic contributions to the magnetizability tensors of several series of molecules and the reader is referred to those papers for details. Especially noteworthy from these studies is that for the normal alkane series, the QTAIM isotropic magnetic susceptibilities of the methylene and methyl groups matched the transferable behavior of other properties for these groups and the magnetic susceptibility of the transferable methyl and methylene groups of the series matched those of Pascal [22]. Also noteworthy was that for the benzene molecule it was found that the threefold increase in magnetic susceptibility for a field applied perpendicular to the plane of the ring, compared with that for a field applied parallel to the plane of the ring, was largely because of the wcðWÞ contributions from the carbon atoms, i.e. the flow of current between the carbon atoms, thus providing a fundamental, physical justification for the famous ring current model of benzene. The significance of this study should not be overlooked. The validity of the ring-current model as an explanation for the magnetic response properties of benzene and other aromatic systems has been debated for decades, usually in terms of orbital models [23]. Keith and Bader analyzed the validity of the ring current model in the only physically reasonable way possible – by actually identifying the atoms in benzene and quantifying the contribution to the magnetizability tensor in terms of the flow of total, physical current within the atoms and between the atoms.

As an additional example of this kind, Table 3.5 shows the principal components of wpðWÞ, wcðWÞ, and wðWÞ for the symmetrically unique atoms of naphthalene (Figs 3.15 and 3.16), calculated at coupled–perturbed HF/6-311þþG(2d,2p)// HF/6-311þþG(2d,2p). Also shown are the isotropically averaged contributions

wiso and the contributions to the major anisotropy waniso. From these results it is apparent that the vast majority of the anisotropy of the magnetic susceptibility in

naphthalene is because of the wcðWÞ contributions from the carbon atoms, which is again consistent with the ring-current model. Note that the wcðWÞzz contribution for each of the two fused carbon atoms, C1 and C6, is approximately 50% larger than that for the other carbons, because the fused carbon atoms have three CaC bonds instead of two and it is the flow of current across the CaC surfaces which is responsible for the large wcðWÞzz for the carbon atoms. These numbers are reflected in the current displays shown in Figs 3.15 and 3.16.