how does the ambiguity of the electronic stress tensor influence its ability to serve as bonding...

How Does the Ambiguity of the Electronic Stress TensorInfluence Its Ability to Serve as Bonding Indicator

Kati Finzel

The electronic stress tensor is not uniquely defined. Possible

bonding indicators originating from the quantum stress tensor

may inherit this ambiguity. Based on a general formula of the

stress tensor this ambiguity can be described by an external

parameter k for indicators derived from the scaled trace of the

stress tensor (whereby the scaling function is proportional to

the Thomas–Fermi kinetic energy density). The influence of kis analyzed and the consequences for the representation of

chemical bonding are discussed in detail. It is found that the

scaled trace of the stress tensor may serve as suitable bonding

indicator over a wide range of k values, excluding the value

range between 20.15 and 20.48. Focusing on the eigenvalues

of the stress tensor, it is found that the sign of the eigenvalues

heavily depends on the chosen representation of the stress

tensor. Therefore, chemical bonding analyses which are based

on the interpretation of the eigenvalue sign (e.g., the spindle

structure) are strongly dependent on the chosen form of the

stress tensor. VC 2014 Wiley Periodicals, Inc.

DOI: 10.1002/qua.24618

Introduction

Quantum chemical topology[1] has gained much interest as its

major task to elucidate the nature of the chemical bond with

the help of real space indicators is at the same time promising

and challenging. One of the most popular approaches in this

field is the Quantum Theory of Atoms in Molecules (QTAIM),[2]

where the electron density serves as basis to define atomic

basins, bond critical points, and molecular graphs. QTAIM

basins may also serve as suitable partitioning schemes, for

example, for the energy decomposition.[3,4] The ratio between

the density Laplacian and the distance to a reference position

defines the local source,[5] whereby the integral of the local

source over a chosen region such as, for example, QTAIM

basins yields the source function. Other bonding indicators,

which may be derived from the electron density are the local

wave vector,[6,7] the one-electron potential,[8,9] and the charge

sampling functional.[10,11] Yet, another bundle of indicators is

based on the pair density like the electron localizability indica-

tor,[12] the spin-pair composition,[13] or the electron localization

function (ELF).[14] ELF can also be seen as based on the kinetic

energy density,[15] a quantity that has also been used to study

chemical bonding.[16,17] The following study is focused on indi-

cators derived from the electronic stress tensor. Whereas the

eigenvalues and the corresponding eigenvectors of the elec-

tronic stress tensor have been intensively studied in relation to

chemical bonding[18–20] and even related to pressure-induced

structural changes,[21] the application as bonding indicator for

the scaled trace of the stress tensor has been proposed with-

out any further examples.[22]

Conversely, the stress tensor is not uniquely defined.[23] Con-

sequently, functions based on the stress tensor may inherit

this ambiguity. The problem is not the use of plurivalently

defined objects for the analysis of the chemical bond—but

the insufficient knowledge about the influence of the induced

ambiguity. Moreover, if its influence is known, the knowledge

could be exploited to extract the most favorable definition of

the electronic stress tensor for the pursuit questions. In the

following, a detailed analysis how the ambiguity of the elec-

tronic stress tensor influences its ability to serve as bonding

indicator is presented for the bonding analysis via the scaled

trace and the bonding analysis via eigenvalues of the stress

tensor. In the first part of next paragraph, the nonuniqueness

of the electronic stress tensor is sketched briefly and the

resulting consequences for the scaled trace and the eigenval-

ues are discussed in part two and three, respectively. There-

after, the implication for the bonding analysis by both

methods is discussed in detail in Results and Discussion.

Theory

For the electronic stress tensor, several definitions exist in liter-

ature.[23–27] This ambiguity can be viewed as rooted in the

indetermination of reducing the Wigner distribution to a set of

position and momentum probabilities, respectively.[28] In other

words, the ambiguity results from the freedom of choice for

the quantum operators. There has been an attempt to clarify

the degree of freedom for the stress tensor. Whereas Godfrey

based his arguments on the gauge freedom,[26] Rogers and

Rappe used a metric argument for their derivation.[27] How-

ever, the resulting expressions for the stress tensor do not

coincide. Therefore, a general formula including two real

parameters a and b has been proposed[23]:

K. Finzel

Max-Planck-Institut f€ur Chemische Physik fester Stoffe, N€othnitzer Str. 40,

Dresden 01187, Germany

E-mail: [email protected]

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r$a;bð~rÞh i

ij52

1

2a@2cð~r ;~r 0Þ@ri@r0j

1@2cð~r ;~r 0Þ@r0i@rj

!"

2ð12aÞ @2cð~r ;~r 0Þ@ri@rj

1@2cð~r ;~r 0Þ@r0i@r0j

!1dijbr2qð~rÞ

#~r5~r 0

:

(1)

Here

cð~r ;~r 0Þ5X

k

nk/�kð~r0Þ/kð~rÞ (2)

is the one-electron reduced density matrix built from the natu-

ral orbitals /kð~rÞ with occupation numbers nk and r15x; r25y;

r35z is shorthand for the Cartesian coordinates, dij is the Kro-

necker delta, and qð~rÞ5cð~r ;~rÞ is the electron density. In case

of real orbitals, the electronic stress tensor as given by Eq. (1)

reduces to:

r$a;bð~rÞh i

ij52 a

Xk

nk@/kð~rÞ@ri

@/kð~rÞ@rj

"

2ð12aÞX

k

nk/kð~rÞ@2/kð~rÞ@ri@rj

11

2dijbr2qð~rÞ

#;

(3)

revealing that r$1;0ð~rÞ is negative semidefinite. This can be

derived from the explicit definition of a negative semidefinite

matrix. For a vector ~x with components ðx1; x2; x3Þ, the scalar

product can be written as:

~x Tr$1;0~x52X

k

nk x1@/kð~rÞ@x

1x2@/kð~rÞ@y

1x3@/kð~rÞ@z

� �2

� 0: (4)

Therefore, the stress tensor r$1;0ð~rÞ can only have negative

or zero eigenvalues.

From Eq. (3), it can be inferred how a and b change the

eigenvalues of the stress tensor. Whereas a changes the eigen-

values of r$a;bð~rÞ and their pairwise differences, b changes the

eigenvalues by a constant shift. A detailed study of the eigen-

value dependence on the parameter set is found in Results

and Discussion part two.

The trace of the stress tensor is proportional to the kinetic

energy density tkð~rÞ:

21

2Tr ½r$a;bð~rÞ�5tkð~rÞ5t1ð~rÞ1kr2qð~rÞ; (5)

which is dependent on the parameter k52ð12a23bÞ=4. The

kinetic energy density tkð~rÞ consists of two parts:

t1ð~rÞ51

2

Xk

nkr/�kð~rÞr/kð~rÞ (6)

the positive definite kinetic energy density and the Laplacian

term kr2qð~rÞ yielding positive and negative contributions.

Whereas the elements of the stress tensor depend on two

parameters a and b, the trace effectively depends only on one

parameter. Thus, the ambiguity of the stress tensor as given

by Eq. (1) produces a relatively simple dependence in tkð~rÞbeing the amount of admixture of the density Laplacian to the

positive kinetic energy density.

Following the concept of Ref. [22], the ability of the scaled

kinetic energy density (whereby the scaling function is propor-

tional to the Thomas–Fermi kinetic energy density) to serve as

bonding indicator is investigated:

tskð~rÞ5

tkð~rÞq5=3ð~rÞ : (7)

Such scaling is frequently used in literature,[14,16] but its effects

are rarely discussed.[29] The influence of the scaling function on

tkð~rÞ and on the ability to show the atomic shell structure has

already been studied elsewhere.[30] It was found that the repre-

sentation of the atomic shell structure for functions of the type

tskð~rÞ is quite robust with respect to changes in k. Functions of

the type tskð~rÞ reveal the atomic shell structure for all k values

above 20.15 and for all k values below 20.25, whereby the

best qualitative behavior (revealing the almost ideal shell occu-

pations) for all atoms from Li to Xe is found for k values around

21/4. Within this study, the effect of k on the tskð~rÞ bonding rep-

resentation for molecules is explored, whereby the scalar field of

tskð~rÞ is expected to provide basins in accordance with chemical

concepts like the Lewis picture. For that reason, the critical

points of tskð~rÞ are explored for different k values:

rtskð~rÞ50: (8)

A critical point in tskð~rÞ is found where the conditions:

05@

@x

t1ð~rÞq5=3ð~rÞ1kx

@

@x

r2qð~rÞq5=3ð~rÞ

05@

@y

t1ð~rÞq5=3ð~rÞ1ky

@

@y

r2qð~rÞq5=3ð~rÞ

05@

@z

t1ð~rÞq5=3ð~rÞ1kz

@

@z

r2qð~rÞq5=3ð~rÞ

(9)

can be fulfilled for one single kscp5ks

x5ksy5ks

z :

kscpð~r cpÞ52

@@i t1ð~r cpÞ2 5

3

@@iqð~r cpÞqð~r cpÞ t1ð~r cpÞ

@@ir2qð~r cpÞ2 5

3

@@iqð~r cpÞqð~r cpÞ r

2qð~r cpÞ(10)

for any choice of direction i. The critical points are further

characterized by its signature. The graph of kscpð~r cpÞ together

with the signature of the critical points provides the basis for

analyzing tskð~rÞ for a given system for different k values in one

single diagram. The function kscpð~r cpÞ yields the values of k for

which tskð~rÞ has a critical point at the position ~r cp, being a

maximum (A), minimum (M), ring critical point (R), or saddle

point (S), respectively. A detailed analysis for the nitrogen mol-

ecule is given in the first part of Results and Discussion.

Computational Details

The nitrogen molecule was calculated with the ADF pro-

gram[31] at Hartree–Fock level with the QZ4P basis at the

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internuclear distance of 2.0136 bohr. Properties were calcu-

lated with a modified DGrid version,[32] where the stress tensor

was implemented according to Eq. (3). The promolecular den-

sity for the nitrogen molecule was calculated at the same

internuclear distance from the atomic HF wavefunctions of

Clementi and Roetti.[33]

Results and Discussion

Bonding analysis via the scaled trace of the stress tensor

Figure 1 displays kscpð~r cpÞ and the signature of the critical points

for the corresponding function tskð~r cpÞ with k5ks

cp using the

converged density (ADF calculation) and the promolecular

density, respectively, for the nitrogen molecule. The function kscp

ð~r cpÞ yields the k value for which the function tskð~rÞ has a critical

point at the desired position ~r cp. The signature sig ðtskÞ charac-

terizes this critical point, being either an attractor (sig ðtskÞ523),

a saddle point (sig ðtskÞ521), a ring critical point (sig ðts

kÞ511),

or a minimum (sig ðtskÞ513). A chosen k value for the function

tskð~rÞ would be represented by a horizontal line in Figure 1.

Then, the intersection with the graph kcpð~r cpÞ yields all posi-

tions at which tskð~rÞ has critical points for that k value. Figure 1

refers to half of the nitrogen molecule, whereby the nitrogen

nucleus is located at~r N51:0068 bohr and the bond midpoint is

at ~r m50 bohr . At both positions ~r N and ~r m, the function tskð~rÞ

has a critical point for any k value, see the vertical lines in

Figure 1 at those positions. At the nucleus, tskð~rÞ will exhibit a

minimum (M) for positive k value and a maximum (A) for nega-

tive k value, respectively, because the behavior of the function

tsk at the atomic nucleus:

tskð~r NÞ5

t1ð~r NÞ1kr2qð~r NÞq5=3ð~r NÞ

(11)

follows the behavior of the density Laplacian approaching

21.[34] At the bond midpoint, the eigenvectors of the Hessian

of ts0 (for k50) point in the same direction as the eigenvectors

of the Hessian of r2q=q5=3. Therefore, the eigenvalues hii of

the Hessian of tsk can be directly deduced from the eigenval-

ues of the above mentioned Hessians:

hiiðtskÞ5hiiðts

0Þ1khiir2q

q5=3

� �; (12)

The eigenvalues hii for the Hessian of the functions ts0 and

r2q=q5=3 at the bond midpoint for the nitrogen molecule are

compiled in Table 1, whereby the molecular axis is along z-

direction. k6 is the value at which the eigenvalue hiiðtskÞ

changes its sign. In the case of the ADF density tskð~rÞ exhibits

a ring critical point (R) at the bond midpoint for k values

larger than 0.25. For k values between 0.25 and 20.14 the

function exhibits a minimum (M) and for k values below

20.14 the function tskð~rÞ has a saddle point (S), as schemati-

cally drawn in Figure 2. For the promolecular density, the kvalues differ, but the qualitative picture is the same.

Figure 1 allows to analyze the bonding description along

the molecular axis for the indicators of the type tskð~rÞ in one

diagram and to compare the descriptions based on promolec-

ular versus converged densities. The position range from ~r cp5

3:0 bohr up to ~r cp51:6 bohr is assigned to the lone pair

region in correspondence to usual chemical concepts as the

Lewis structure. The range from ~r cp51:6 up to ~r cp51:3 bohr

represents the outer border (next to the lone pair region) of

the nitrogen core, whereby the core region range from ~r cp5

1:3 to ~r cp50:7 bohr . Position values between ~r cp50:7 and ~r cp

50:4 bohr represent the inner border (next to the bond mid-

point) of the nitrogen core and values from ~r cp50:4 up to ~r cp

50:0 bohr display the bonding region.

The data from Figure 1 are recompiled in Tables 2 and 3 for

convenience. Table 2 contains the possible tsk bonding

Figure 1. kscpð~r cpÞ and signature of ts

kð~rÞ for the corresponding k5kcpð~r cpÞ for

the nitrogen molecule along the molecular axis. The bond midpoint is situ-

ated at r50:0 bohr. Black: kscpð~r cpÞ for the promolecular density, red: ks

cpð~r cpÞfor the ADF density, gray: signature of ts

kð~rÞ for the promolecular density, and

orange: signature of tskð~rÞ for the ADF density. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]

Table 1. Eigenvalues of the Hessian of ts0 and r

2q

q5=3 perpendicular (x-direc-

tion) and along the molecular axis (z-direction) at the bond midpoint for

the nitrogen molecule at equilibrium distance.

Density Component ts0

r2qq5=3 k6

ADF hxx 2.1 15.2 20.14

hzz 11.1 244.4 0.25

Promolecular hxx 1.3 14.4 20.09

hzz 7.9 280.5 0.10

k6 is the value at which the eigenvalue of the Hessian of tsk changes its

sign.

Figure 2. Signature for the critical point of tsk at the bond midpoint for dif-

ferent k values for the ADF density. R stands for ring critical point, M for

minimum, and S for saddle point, respectively. The results for the promo-

lecular density are in parentheses.



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descriptions for the converged density. As can be seen from

the first row of Table 2, for k values larger than 0.25, the func-

tion tskð~rÞ shows a minimum (M) in the lone pair region, a ring

critical point (R) at the outer core boundary, a minimum (M) at

the nitrogen nucleus, a ring critical point (R) at the inner core

boundary and two minima separated by a ring critical point in

the bonding region (MRM, i.e. minimum, ring critical point,

minimum), whereby the ring critical point is located at the

bond midpoint. For that reason only one of the minima and

the ring critical point is noted in Tables 2 and 3, because the

other minimum is located on the negative axis.

Consequently, inverting the topology for tskð~rÞ, for example,

by taking 2tskð~rÞ, yields for k values larger than 0.25 an attrac-

tor (A) in the lone pair region, a saddle point (S) on either side

of the nitrogen core, one attractor (A) for each nitrogen core,

and two attractors (A) in the bonding region separated by a

saddle point (S). In short, this reads (A S A S AS) which can be

seen as the inverse of the first row of Table 2. Thus, functions

of the type 2tskð~rÞ for k � 0:25 show separate basins for the

lone pair regions, for the nitrogen cores and two basins in the

bonding region.

For k values between 0.24 and 20.14, see second and third

row of Table 2, the function tskð~rÞ exhibits one single minimum

(M) in the bonding region and thus, the bonding region is

described by one single basin (for attractor A) if functions of the

type 2tskð~rÞ are used as bonding indicators. For k values

between 20.01 and 20.14 three critical points are found in the

core region (as a horizontal line for specific k in this range inter-

sects kscpð~r cpÞ three times within the core region). At the position

of the nucleus tskð~rÞ exhibits a maximum (A). Next to the nucleus,

the function tskð~rÞ shows a saddle point (S) and a minimum (M),

respectively. Consequently, inverting the topology (by taking

2tskð~rÞ) yields (R M A) in the core region (which is the opposite

of the third row in Table 2) and thus, only one basin, namely for

the attractor (A), is found in the core region if functions of the

type 2tskð~rÞ are used as bonding indicators. However, the attrac-

tor of this basin does not coincide with the nitrogen nucleus.

k values between 20.15 and 20.24 have been excluded

from the previous investigation, because for this k range the

function tskð~rÞ does not display proper shell boundaries.[30]

k values between 20.25 and 20.48, see fifth row of Table 2,

do not produce suitable bonding indicators, as for those k val-

ues the function tskð~rÞ does not exhibit attractors in the bond-

ing region and thus, bonding basins are missing. For that

reason, the function ts21=4ð~rÞ, which yields the best atomic

shell representation,[30] is not a suitable bonding indicator, as

it does not show a bonding basin for the nitrogen molecule.

Instead, the function tskð~rÞ with k values below 20.49, see

sixth and seventh row of Table 2, may serve as bonding indi-

cator, with tskð~rÞ exhibiting an attractor at the atomic nucleus,

two attractors in the bonding region and for the lone pair

region either a ring attractor for values between 20.49 and

20.59 or one single attractor on the molecular axis for k val-

ues below 20.60. All k values below 20.49 will lead to a

description with tskð~rÞ showing separate lone pair basins, core

basins for each nucleus, and two basins in the bonding region.

For converged densities, the functions of the type tskð~rÞ do

not show attractors in the mirror plane perpendicular to the

molecular axis running through the bond midpoint. This can

be seen in Figure 3, where kscpð~r cpÞ perpendicular to the

molecular axis is plotted. The values for kscpð~r cpÞ vary only

between 20.15 and 20.19 (that range was already excluded

as it does not reveal the proper core radii) and thus, outside

this range tskð~rÞ does exhibit critical points along this direction.

Therefore, all possible bonding indicators of type 6tskð~rÞ for k

values above 20.14 and below 20.49 do not indicate a ring

attractor (which could be interpreted as originating from the

triple bond) perpendicular to the molecular axis.

The data in Table 2 shows that tskð~rÞ may serve as suitable

bonding indicator over a wide range of k values, namely for kvalues above 20.14, see first, second, and third row, and for kvalues below 20.49, see sixth and seventh row of Table 2.

Table 3 compiles the bonding description for tskð~rÞ based on

the promolecular density for different k ranges. Only k values

between 2.14 and 0.13 lead to a chemical intuitive description

of the system. For all other k values, an attractor for either

lone pair or bonding region is missing. This shows that tskð~rÞ

distinguishes between the promolecular and the converged

density (except for k values between 2.14 and 0.25).

For k values above 20.14, the function 2tskð~rÞ and for k val-

ues below 20.49, the function tskð~rÞ may serve as suitable

bonding indicators, as all corresponding functions display one

attractor in the lone pair region, one attractor for the core

Table 2. Signature of critical points of tsk along the molecular axis of the

nitrogen molecule for different k ranges calculated from the ADF density.

Lone pair Outer border Core Inner border Bond k range

M R M R MR ½11; 0:25�M R M R M [0.24, 0.00]

M R SAM S M [20.01, 20.14]

k range excluded from the investigation[a] [20.15, 20.24]

R[b] S A S RS [20.25, 20.48]

R[b] S A S AS [20.49, 20.59]

A S A S AS [20.60, 21]

[a] Within this range, the inner shell boundaries are not properly

described.[30] [b] Ring critical point along the molecular axis accompa-

nied by a ring attractor in the molecular plane.

Table 3. Signature of critical points of tsk along the molecular axis of the

nitrogen molecule for different k ranges calculated from the promolecu-

lar density.

Lone pair Outer border Core Inner border Bond k range

M A M R SR ½11; 2:15�M A M R MR ½2:14; 0:13�S A M R MR ½0:12; 0:10�S A M R M ½0:09; 0:00�S A SAM R M ½20:01;20:09�S A SAM R S[a] ½20:10;20:14�k range excluded from the investigation[b] ½20:15;20:24�A M A S RS ½20:25;21�

[a] Saddle point accompanied by a ring minimum in the mirror plane

perpendicular to the molecular axis. [b] Within this range, the inner

shell boundaries are not properly described.[30]



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region, and one or two attractors in the bonding region. Addi-

tionally, the basin populations for different k have been eval-

uated. The data are collected in Table 4. The population of the

core basin does not change significantly for different k, being

roughly 2.0 electrons for all k values above 20.14 or below

20.49. The population of the bonding basin is compiled in the

last column of Table 4, whereby the bonding indicator may

either exhibit one or two basins in the bonding region

depending on the choice of k. The value k520:49, see first

row of Table 4, yields the smallest basin population of 2.8 elec-

trons for the bonding basin of the nitrogen molecule. The

largest bonding basin population of 4.9 electrons is reached

with k520:14, see last row of Table 4. The dependence of the

bonding population on the k value can be reasoned from Fig-

ure 4, where the saddle points between the bonding basin

and the lone pair basin for the nitrogen molecule for different

k values are depicted. All saddle points between the bonding

and the lone pair basin lie on the black line (upper part of the

figure). On this line, Eqs. (9) are fulfilled for all three directions

for the same k value. The colored areas show the value for kz

calculated according to Eq. (11). The kz value can of course be

evaluated anywhere in space, but only at the black line kx5ky

5kz which is the condition for the occurrence of a critical

point for tskð~rÞ.

Within the k range that is suitable for chemical description,

the value k520:49 yields the smallest possible bonding popu-

lation. The saddle point of tskð~rÞ for k520:49 is situated in the

violet area of Figure 4 close to the black area, which repre-

sents the k range that was excluded, because for those krange, the function ts

kð~rÞ does not reveal bonding basins. The

basin boundary between the lone pair and the bonding basin

runs through the saddle point (S). From Figure 4, it can be

rationalized that tskð~rÞ for k520:49 provides the smallest

region for the bonding basin, by assuming the basin bounda-

ries between lone pair and bonding basin to be more or less

straight lines in the depicted molecular plane, running through

the nitrogen nucleus and the corresponding saddle point.

Then, the size of the bonding basin increases with increasing

angle between the molecular axis assumed basin boundary.

Choosing k values lower than 20.49 successively augments

the population of the bonding basin (going on the black line

from 20.49 upward to 21) until it reaches 3.8 electrons for

the function ts21ð~rÞ, which is proportional to the negative

scaled density Laplacian 2r2q=q5=3. For k!1, the function

2tskð~rÞ is analyzed, yielding of course the same basin popula-

tions as for the negative scaled density Laplacian. Decreasing

the k value (following the black line upward from 11) succes-

sively augments the basin population of the bonding basin for

functions of the type 2tskð~rÞ. The function 2ts

0ð~rÞ, which has

the same topology as the localized orbital locator, yields a

bonding description with one single bonding basin containing

4.3 electrons. The corresponding saddle point is located at the

border between the green and the dark green area. But still

higher bonding populations may be reached by choosing

Figure 3. kscpð~r cpÞ and the signature of ts

kð~rÞ for the corresponding k5kð~r cpÞin the mirror plane perpendicular to the molecular axis for the nitrogen mol-

ecule. The bond midpoint is situated at r50:0 bohr . Black: kscpð~r cpÞ for the

promolecular density, red: kscpð~r cpÞ for the ADF density, gray: signature of

tskð~r cpÞ for the promolecular density, and orange: signature of ts

kð~r cpÞ for the

ADF density. [Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]

Table 4. Basin population for the nitrogen molecule for different kvalues.

k Function Lone pair Core Bond

20.49 1 3.6 1.9 1.4 1 1.4 5 2.8

21 1 3.1 2.0 1.9 1 1.9 5 3.8

11 2 3.1 2.0 1.9 1 1.9 5 3.8

0.00 2 2.8 2.0 4.3

20.14 2 2.5 2.0 4.9

Function indicates which type of function 6tskð~rÞ has been used.

Figure 4. Schematic representation how the population of the bonding

basin depends on the chosen k value for the nitrogen molecule. cyan: core

basin of the nitrogen nucleus. The bond midpoint is on the right part of

the figure (yellow colored region). Saddle points of tskð~rÞ for

k520:14; 0:00;61, and 20.49 between lone pair basin and bonding

basin are depicted in red. All saddle points lie on the black line. For those

points the Eqs. (9) are fulfilled for all three directions. The colored areas

represent kscp for the z-direction according to Eq. (11). Violet area: k from

21 to 20.45, black area: k from 20.45 to 20.15, dark green area: k from

20.15 to 0.00, light green area: k from 0.00 to 0.15, light orange area: kfrom 0.15 to 0.30, dark orange area: k from 0.30 to 0.45, lavender area: kfrom 0.45 to 0.60, and white area: k from 0.60 to 11. [Color figure can be

viewed in the online issue, which is available at wileyonlinelibrary.com.]





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lower k values (successively following upward on the black

line). The largest population of 4.9 electrons for the bonding

basin is obtained for k520:14. The saddle point for 2ts20:14ð~rÞ

is located at the boundary between the black area and the

dark green area in Figure 4, whereby the black area represents

the excluded k range. Therefore, the function 2ts20:14ð~rÞ yields

the bonding description for the nitrogen molecule with the

largest bonding basin population that can be reached within

that function pool.

The quality of the bonding description as shown by func-

tions of the type 6tskð~rÞ is collected in Figure 5. Functions of

the type 1tskð~rÞ may serve as suitable bonding indicators for k

values below 20.49, white area in the lower part of the figure.

The description yields in all cases two bonding basins. Func-

tions of the type 2tskð~rÞ may also serve as suitable bonding

indicators for k values above 20.14, see the white and dotted

area in the upper part of Figure 5. The description yields one

single bonding basin for k values up to 0.25. For larger k val-

ues, two bonding basins are found. The largest electron popu-

lation of 4.9 electrons for the bonding basin is found for

k520:14, represented by the green line. Despite the satisfac-

tory results concerning the representation of the atomic shell

structure[30] the function ts21=4ð~rÞ must be excluded from the

set of possible bonding indicators, because its topology does

not reveal a bonding region for the nitrogen molecule. If the

bonding indicator shall distinguish between the topology of

the promolecular and the topology of the converged density,

values between 2.14 and 0.25 have to be excluded as well.

Bonding analysis via stress tensor eigenvectors and

eigenvalues

Stress tensor eigenvectors and the corresponding eigenvalues

were in certain cases used to elucidate chemical bonding[18]

and structural changes of molecules.[21] Both analysis refer to

the tensile (positive eigenvalues) and the compressive (nega-

tive eigenvalues) modes of a volume element, deducing from

that analysis the most favorable way of electron density

changes. However, as the stress tensor is not uniquely given,

the tensile mode may change into a compressive mode for dif-

ferent representations of the stress tensor. Therefore, this sec-

tion is devoted to the influence of the chosen stress tensor

form on the principal electronic stresses.

For approximative stress tensors obtained via gradient

expansion, it has been found that the eigenvectors of such

approximative stress tensors at a bond critical point are close

to the eigenvectors of the electron density Hessian matrix, irre-

spective of the chosen stress tensor representation.[21]

By examining the off diagonal elements of the electronic

stress tensor [Eq. (3)]:

r$a;bð~rÞh i

ij52 a

Xk

nk@/kð~rÞ@ri

@/kð~rÞ@rj

"

2ð12aÞX

k


#

52a2

@2

@ri@rjqð~rÞ1

Xk


(13)

it can be seen that this is also true for the exact stress tensor

if the absolute value for a is large enough. Moreover, b has no

influence on the stress tensor eigenvectors, because the off

diagonal elements of the stress tensor do not depend on the

b value. However, the stress tensor eigenvalues heavily depend

on the chosen representation. Therefore, the eigenvalues of

the electron density Hessian cannot serve as suitable approxi-

mations to the exact stress tensor eigenvalues as will be

shown in the next paragraph.

Influence on the eigenvalues. At a bond critical point the

electron density Hessian has signature 21, having one positive

eigenvalue along the bond direction and two negative eigen-

values perpendicular to it. Those eigenvalues can only serve as

approximations to the stress tensor eigenvalues if the stress

tensor signature at the bond critical point is also 21. However,

stress tensors as given by Eq. (3) do not necessarily have that

signature at the bond critical point. Depending on the choice

of a and b, the stress tensor signature at a bond critical point

can have any possible value. This will be demonstrated for the

nitrogen molecule, whereby the molecular axis is oriented

along z-direction.

Due to symmetry, the eigenvalues r$a;bð~rÞh i

iiof a dimer at

the bond critical point can be evaluated directly from the

stress tensor Eq. (3):

r$a;bð~rÞh i

ii522t1ið~rÞ1

12a2r2

i qð~rÞ21

2br2qð~rÞ; (14)

where 2t1ið~rÞ5X

k

nkð@/kð~rÞ=@riÞ2 is the component of the

positive kinetic energy density along i-direction and the term

Figure 5. kscpð~r cpÞ for the nitrogen molecule along the molecular axis. The

bond midpoint is situated at r50:0 bohr . Red: kscpð~r cpÞ for the ADF density,

doubly dashed area: k range not suitable for description of core bounda-

ries, dashed area: k range not suitable for description of bonding region,

dotted area: beginning at k52:14, the function tskð~rÞ yields the same topo-

logical description for the promolecular and the converged density, green:

k520:14 yields the largest population (of 4.9 electrons) for the bonding

basin. [Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]




http://q-chem.org/


r2i qð~rÞ5@2qð~rÞ=@r2

i is the second derivative of the density

along the i-direction. For a51;b50, the eigenvalues are given

directly by t1ið~rÞ and, therefore, are always negative, see also

Eqs. (3) and (4). For large absolute values of a and moderate bvalues, the eigenvalues of the stress tensor are governed by

the corresponding second derivative of the density. For such

stress tensors, the eigenvalues of the electronic density Hes-

sian may serve as approximations for the stress tensor eigen-

values. However, for large absolute values of b and moderate

a values, the eigenvalues of the stress tensor are of the same

sign, governed by the Laplacian term, see Eq. (14). In such

cases, the stress tensor signature at the bond critical point

does not equal the signature of the electron density Hessian

and, therefore, the eigenvalues of one matrix cannot serve as

suitable approximations for the eigenvalues of the second

matrix.

In Figures 6 and 7, the signature of the electronic stress ten-

sor is shown as a function of a for two qualitative different

choices of b. a6i is the a value for which the stress tensor

eigenvalue r$a;bð~rÞh i

iichanges its sign:

a6i 52

4t1i

r2qi

112br2qr2

i q: (15)

a6i depends on the parameter b. The choice of b influences

the stress tensor signature in so far, as it determines the order-

ing of a6i . The value b0 for which all eigenvalues change their

sign at the same a6 is given by:

b054r2qx t1z24r2qzt1x

r2qr2qz2r2qr2qx

: (16)

For such a choice of b, the stress tensor signature at the

bond critical point is 61 depending on the choice of a. Choos-

ing b below b0 yields a6z < a6

x . For that case, the stress tensor

signature at the bond critical point is given in Figure 6. For abelow a6

z , the signature will be 21. For such stress tensors,

the signature at the bond critical point is same as the signa-

ture of the electron density Hessian, compare also Eq. (14).

However, choosing a above a6z changes the sign of the eigen-

value for the z-direction and thus, all eigenvalues are negative

until a increases above a6x . For such stress tensor representa-

tion, the signature at the bond critical point is 11.

In Figure 7, the stress tensor signature is shown as a func-

tion of a for b > b0. In that case a6x < a6

z and, therefore, the

stress tensor signature is 13 if a lies between a6x and a6

z .

The above analysis shows that the stress tensor eigenvalues

at the bond critical point can be of any sign depending on

the stress tensor representation.

Most common choices for the stress tensor are restricted to:

0 � a13b � 1 (17)

or even more restrictive with b50, leading to:

0 � a � 1: (18)

In the case of r$a;0 stress tensors, only those eigenvalues

may change sign, for which the second density derivative is

positive, compare Eq. (16), because the sign of the eigenvalue

can only change if a6i 5½0; 1� and t1i is always positive. For a

dimer, this is the eigenvalue for which the eigenvector is ori-

ented along the bond direction. The remaining eigenvalues

(for which the eigenvectors are perpendicular to the molecular

axis) are negative for any a5½0; 1�. Therefore, the signature of

stress tensors r$a;0 can be either 21 or 23 for a5½0; 1�. This is

schematically shown in Figure 8.

The value a6z depends on the ratio between t1z and r2qz .

a6z lies between 0 and 1 if 4 t1z � r2qz, which is fulfilled at

the bond critical point of the nitrogen molecule. Thus, there

must be a change of sign of the eigenvalue on the interval

a5½0; 1�, which means that different stress tensor representa-

tions give different answers about magnitude and sign of the

eigenvalue (for which the eigenvector is oriented along the

bond direction) which in some cases has been connected with

the strength of the bond.[21] Only regions in which the part of

the kinetic energy along the bond is high compared to the

second density derivative along that direction (4t1z > r2qz)

do not suffer from the ambiguity of r$a;0 stress tensors. In

those regions, all eigenvalues are negative for any representa-

tion of the stress tensor r$a;0 with a5½0; 1�.Releasing the restriction b50, the stress tensor signature

may have any possible value again. As there always exists a

Figure 6. Signature of the electronic stress tensor at the bond critical point

for a dimer oriented along z-direction as a function of a if b is chosen

below b0.


for a dimer oriented along z-direction as a function of a if b is chosen

above b0.



http://q-chem.org/




corresponding b for every a value which shifts the sum Eq.

(17) between 0 and 1 for such a value for which the eigenval-

ues, whose eigenvectors are perpendicular to the molecular

axis, are positive. Therefore, if the general formula for the

stress tensor is given by Eq. (1) and there are no further moti-

vated choices for a and b, the sign and magnitude of the

eigenvalues are strongly dependent on the specific choice of

the stress tensor, affecting the interpretation of the eigenval-

ues sign (compressive/tensile modes).

Influence on the spindle structure. As shown in the previous

paragraph, the eigenvalues heavily depend on the stress ten-

sor representation. For that reason, the spindle structure also

depends on the specific form of the stress tensor. The spindle

structure is a region of positive stress tensor eigenvalues con-

necting a pair of regions RD of atoms, whereby the region RD

of an atom is the region of positive kinetic energy density

around the atomic nucleus.[18] The region RD depends of

course on the representation of the kinetic energy density. For

k50 (corresponding, e.g., to a51;b50), the region RD is infi-

nite as the kinetic energy density arising from this stress ten-

sor t0ð~rÞ is positive everywhere. For k!1, the kinetic energy

density is dominated by the density Laplacian and the region

RD will be very close to the first shell of density accumulation

as given by the electronic density Laplacian.

However, the region of space where the stress tensor has

positive eigenvalues also depends on its chosen representa-

tion. In Figure 9, the largest eigenvalue of the stress tensor

r$a;0 is plotted for the nitrogen molecule using different a val-

ues. For a50, a large region with positive eigenvalues is found

in between the two nitrogen nuclei, see yellow colored region

in Figure 9a. For a51=2 that region, having positive stress ten-

sor eigenvalues, is smaller but still visible, see Figure 9b. In Fig-

ure 9c, the largest eigenvalue is plotted for r$1;0. This stress

tensor form has no positive eigenvalues, compare also Eqs. (3)

and (4) and thus, this form cannot exhibit a spindle structure

for the nitrogen molecule.

Conclusions

In this study, it has been shown how the nonuniqueness of

the electronic stress tensor affects its ability to serve as bond-

ing indicator. The first part of this study is focused on possible

bonding indicators derived from the scaled trace of the stress

tensor. It has been found that functions of the type 2tskð~rÞ

may serve as suitable bonding indicators for k values larger

than 20.14, whereby the most chemical appealing bonding

description for the nitrogen molecule is given by the function

2ts20:14, for which the bonding basin has the highest possible

electron population (of 4.9 electrons) among the functions of

type 6tskð~rÞ. Functions of the type ts

kð~rÞ may also serve as pos-

sible bonding indicators if k is chosen below 20.49. All indica-

tors of this type show two basins in the bonding region (this

is also the case for 2tskð~rÞ for k values above 0.25). Indicators

of type 6tskð~rÞ differentiate between the topology of the con-

verged and the topology of the promolecular density, except

on the set of k values between 2.14 and 0.25. The function


for a dimer oriented along z-direction as a function of a5½0; 1� for b50

and 4t1z � r2z q. If 4t1z > r2

z q, then a6z < 0 and the signature of the elec-

tronic stress tensor is 23 for all values a5½0; 1�. a6x is larger than 1. There-

fore, the eigenvalues, whose eigenvectors are perpendicular to the bond

direction, may not change sign for a5½0; 1�.

Figure 9. Largest eigenvalue of the stress tensor r$a;0 for different a values

for the nitrogen molecule. [Color figure can be viewed in the online issue,

which is available at wileyonlinelibrary.com.]




http://q-chem.org/


ts21=4 (despite its appealing performance on atomic data) is

not suitable for bonding analysis, as it does not reveal a bond-

ing basin between the two nitrogen atoms.

In the second part of this study, it was shown that the sign

of the stress tensor eigenvalues depends on its chosen repre-

sentation. For a dimer at the bond critical point, the signature

of the electronic stress tensor can be either 23, 21, 11, or

13 depending on the chosen stress tensor form. Therefore,

the outcome of an analysis which is based on the sign of the

stress tensor eigenvalue (like tensile and compressive modes)

is influenced by the chosen stress tensor representation. Even

for stress tensors of type r$a;0, the eigenvalue for which the

eigenvector is oriented along the bond direction changes sign

for a certain representation on the set a5½0; 1�. That findings

also apply to the spindle structure, where the size of the

region having positive stress tensor eigenvalues heavily

depends on a and b. Moreover, the stress tensor r$1;0 will not

exhibit a spindle structure as its eigenvalues are always

negative.

Acknowledgment

The author gratefully thanks Miroslav Kohout for fruitful discussions

and carefully reading the manuscript.

Keywords: bonding analysis � stress tensor � spindle structure •

kinetic energy density � molecules

How to cite this article: K. Finzel Int. J. Quantum Chem. 2014,

114, 568–576. DOI: 10.1002/qua.24618

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Received: 3 December 2013Revised: 8 January 2014Accepted: 17 January 2014Published online 6 February 2014



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