Commentaries on quantum similarity (1): Density gradient quantum similarity

Commentaries on Quantum Similarity (1): Density

Gradient Quantum Similarity

RAMON CARBO-DORCA, LUZ DARY MERCADO

Institut de Quımica Computacional, Universitat de Girona, Girona 17071, Catalonia, Spain

Received 23 September 2009; Revised 1 December 2009; Accepted 29 December 2009DOI 10.1002/jcc.21510

Published online 24 March 2010 in Wiley InterScience (www.interscience.wiley.com).

Abstract: Computation of density gradient quantum similarity integrals is analyzed, while comparing such inte-

grals with overlap density quantum similarity measures. Gradient quantum similarity corresponds to another kind of

numerical similarity assessment between a pair of molecular frames, which contrarily to the usual up to date quan-

tum similarity definitions are not measures, that is: strictly positive definite integrals. As the density gradient quan-

tum similarity integrals are defined as scalar products of three real functions, they appear to possess a richer struc-

ture than the corresponding positive definite density overlap quantum similarity measures, while preserving the over-

all similarity trends, when the molecular frames are relatively moved in three-dimensional space. Similarity indices

are also studied when simple cases are analyzed in order to perform more comparisons with density overlap quantum

similarity. Multiple gradient quantum similarity integrals are also defined. General GTO formulae are given. Numeri-

cal results within the atomic shell approximation (ASA) framework are presented as simple examples showing the

new performances of the gradient density quantum similarity. Fortran 90 programs illustrating the proposed theoreti-

cal development can be downloaded from appropriate websites.

q 2010 Wiley Periodicals, Inc. J Comput Chem 31: 2195–2212, 2010

Key words: quantum similarity; density gradient quantum similarity integrals; density overlap quantum similarity

measures; quantum similarity indices; ASA framework

Introduction

Preliminary Aims and Considerations

Quantum molecular similarity since the first naıve study on how

similar can be two molecules from a quantum chemical point of

view1 has been steadily developing up to now.2–46 A recent

review47 provides with a broad historical and up to date pano-

rama of the theoretical structure of quantum similarity. The cus-

tomary use of several positive definite operators, weighting den-

sity function products, has been currently considered as the var-

ied building block collection of quantum similarity measures.

Quantum similarity measures have been employed in the theo-

retical setup of Quantum QSPR46–61,142 and used in practical

related questions as well.62–103,143,144 In some instances, differ-

ential operators like the Laplacian have been computationally

implemented, theoretically analyzed104 and included in Quantum

QSPR studies from the theoretical point of view.55,56

However, as Bader has pointed out in another context,105 first

order density functions gradient and Laplacian can hold a funda-

mental role in order to understand atoms in molecules and

chemical bonding. The gradient of density functions has been

also studied in order to discover unnoticed aspects of the density

functions,106 among other characteristics107–110 of such a funda-

mental quantum chemical tool, which developed into a leading

position among modern alternative theoretical studies of molecu-

lar structure.111

Here, in this study, the gradient of the electronic part of the

first order density function is proposed to become a sound mo-

lecular descriptor for quantum similarity measuring purposes.

The present communication pretends to constitute an analytical

and numerical window to compare density gradient similarity

integrals with the well known simpler overlap quantum similar-

ity measures. It is the first of a series of reflections on the nature

of both theoretical and computational aspects of quantum simi-

larity measures, pretending to disclose new facets of the subject.

For this reason, the present work is providing the literature on

the subject developed in this laboratory.

The proposed cycle of papers will develop several aspects,

going from simple theoretical examples to introduce and

enhance the pedagogical side of the problem up to practical

Contract/grant sponsor: Spanish Ministerio de Educacion y Ciencia; con-

tract/grant number: CTQ2006-04410/BQU

Correspondence to: Prof. R. Carbo-Dorca; e-mail: quantumqsar@hot

mail.com

q 2010 Wiley Periodicals, Inc.

algorithms, dealing with real computations on atoms and mole-

cules. Our lemma here and in the following related papers will be

reproducibility, for this reason the examples displayed at the end

of the present study are associated to the corresponding programs

and the needed running information, which can be applied to other

problems the readers consider interesting to further develop. Such

endeavor is also trying to constitute a source of easy practical com-

putational elements, which will be disclosed in public download-

ing websites and that will accompany every issue of the series.

At the same time as the previous generic considerations will

be developed, the present work pretends to provide a novel pos-

sibility to define quantum similarity integrals in a more general

way than as has been done up to date. Indeed, in comparison of

already well-known overlap quantum similarity measures, gradi-

ent similarity integrals which will be disclosed here, are not

sharing in some aspects the same positive definite behavior as

usual quantum similarity measures have. A particular aim of this

article is to show this atypical facet associated to gradient simi-

larity integrals. To shed light in this new feature of quantum

similarity, a parallel comparison of gradient integrals with over-

lap measures will be present in any case example.

To achieve those goals as best as possible, some preliminary

notation after this introduction will initiate an elementary analyt-

ical example presented under Gaussian functions, which will be

followed with a slightly complicated one as a way to establish

the background to construct gradient similarity measures within

ASA density functions.112–119 Some numerical gradient similar-

ity examples in the ASA context will be given next, followed by

a discussion dealing with general Gaussian functions, which will

be presented for the sake of completeness. The integral defini-

tion of hybrid density-gradient density comparisons in form of a

three-dimensional vector and its definite positive scalar Eucli-

dian norm will be also described as a trivial consequence of the

previous discussion. A proposal of multiple quantum similarity

measures involving gradient density functions will complete the

theoretical part of this study. At the end of this article some sim-

ple numerical examples will be presented to compare the behav-

ior of overlap similarity with gradient similarity performances.

Such examples contain atomic and molecular self-similarities

first, then bimolecular superposition studies and diatomic inter-

actions will be studied in order to shed light into the gradient

and overlap similarity differences.

Notation and Mathematical Background

Suppose known a first order density function in a LCAO con-

text, which can be written as:

q rð Þ ¼Xl

Xm

Dlmvl rð Þvm rð Þ (1)

where D 5 {Dlm} is the symmetric charge and bond order ma-

trix and X 5 {vl} a set of monoelectronic basis set real func-

tions, see for example the series of papers on density func-

tions106–110,120 for more notation assistance.

The gradient of the density function (1) is easily computed

taking into account the symmetrical nature of the matrix D, as:

@q rð Þ@r

¼ 2Xl

Xm

Dlmvl rð Þ @vm rð Þ@r

(2)

Then, after this simple description, instead of the density itself

as presented in eq. (1) one can use the gradient (2) for quantum

similarity purposes, even if instead of a scalar, like the density

function is, the gradient appears to be a three-dimensional

vector.

That is, one can define the similarity integral1–46,48–61

between two (or more) molecular structures with densities

described as: {qA, qB} and integrating the dot product of the

corresponding density gradient vectors in the following way:

gAB ¼ @qA@r

�� @qB@r

� �¼

ZD

@qA@r

;@qB@r

� �dr

¼XI

ZD

@qA@xI

@qB@xI

dr ð3Þ

where D is an appropriate integration domain and the sum runs

over the three electron position coordinates: r 5 {xI|I 5 1,3}.

Because of Green’s theorem,121 one can also write:

gAB ¼ZD

@qA@r

;@qB@r

� �dr

¼ �ZD

qA rð ÞD qB rð Þð Þdr ¼ � qADqBh i; ð4Þ

where the symbol: D ¼ PI

@2

@2xIstands for the Laplacian operator.

Defined in the way of eqs. (3) or (4), the integral involving

both gradients or the Laplacian operator cannot be longer associ-

ated to a measure, but to a mathematical construct made of

some kind of a scalar product, formed at the same time by the

scalar products of the gradient components. Such an integral can

be also interpreted as a weighted scalar product of two density

functions as shows the equation earlier, with the Laplacian oper-

ator acting as an integral weight. Alternatively, such a mathe-

matical definition can be also associated to a kinetic energy-like

integral, involving in general the densities of two different mo-

lecular structures, instead of the associated wave functions.

Now there is needed to comment, for the sake of complete-

ness, that such a similarity gradient scalar product can yield any

real number, contrarily to the usual features present in the defini-

tions of the standard similarity integrals, see for instance ref. 16.

Similarity integrals have been customarily built up as positive

definite integrals explicitly, as a result they can be considered

well defined measures. The source of the intrinsic positive definite

feature of usual quantum similarity integrals has to be found,

without doubt, in the positive definite nature of the density

functions. Yet, such a characteristic may be absent in gradients of

density functions, therefore the similarity integrals involving

density gradients may yield real numbers not necessarily positive

definite.

In the same way, as in analogous molecular similarity cases,

one can define gradient self-similarity measures; see for example

ref. 7, for the involved molecular structures appearing in the

2196 Carbo-Dorca and Mercado • Vol. 31, No. 11 • Journal of Computational Chemistry

Journal of Computational Chemistry DOI 10.1002/jcc

similarity integral (4) in terms of the Laplacian weighted Euclid-

ean norms:

gAA ¼ � qADqAh i ^ gBB ¼ � qBDqBh i;

which in this case can be strictly associated to positive definite

real numbers.

The usual similarity and dissimilarity indices, see for exam-

ple refs. 1,11,17, that is, the cosine and Euclidian distance,

within the density gradient similarity framework can be easily

described respectively as:

rAB ¼ gAB gAAgBBð Þ�12^dAB ¼ gAA þ gBB � 2gABð Þ12:

A Simple Example

After having set up the schematic working background and in

order to sketchily describe the nature of the density gradient

similarity integrals, a straightforward example employing just

one Gaussian function will be discussed here as a first step.

Suppose the density function being defined by a unique

squared Gaussian function:

c r��a� � ¼ N 2að Þ exp �2a

��r��2� �; (5)

where:

N 2að Þ ¼ 2ap

8>: 9>;32

is a normalization factor such that the following Minkowski

norm holds for the function (5), that is, it can be formally

written:

c r��a� �� ¼ Z

D

c r��a� �

dr ¼ 1;

with the integration domain D taken over the whole three-

dimensional space. There has been employed the well-known

integral122:

i0 að Þ ¼Z þ1

�1exp �ax2

� �dx ¼ p

a

8: 9;12

:

The gradient of the function (5) is readily computed as:

@c ¼ @c r��a� �

@r¼ �4aN 2að Þ exp �2a

��r��2� �r

and so, the associated gradient self-similarity is easily written by

means of the following integral:

@c að Þj j2D E

¼ 4að Þ2N2 2að ÞZD

rj j2exp �4a rj j2� �

dr; (6)

thus, in order to obtain the gradient norm (6) one shall now also

use the integral122:

i2 að Þ ¼Z þ1

�1x2 exp �ax2

� �dx ¼ 1

2

pa3

8: 9;12

;

which permits to write the norm (6) as the final result:

@c að Þj j2D E

¼ 6aap

8: 9;32

; (7)

and such a norm for any other Gaussian function with different

exponent becomes:

@c bð Þj j2D E

¼ 6bbp

8>: 9>;32

:

Finally, the similarity integral between two Gaussian function

gradients can be written explicitly as:

@c að Þ@c bð Þh i ¼ 242

p3

8>: 9>;12 ab

aþ bð Þ8>>: 9>>;5

2

¼ 62abaþ bð Þ

8>>: 9>>; 2abp aþ bð Þ

8>>: 9>>;32

¼ 6kkp

8>: 9>;32

ð8Þ

with the aid of a new parameter defined like:

k ¼ 2ab

aþ bð Þ (9)

which when both exponents are the same becomes the sole

exponent.

Equation (8) has the same form as the gradient self-similar-

ities shown in eq. (7). Thus, one can formally write:

@c að Þ@c bð Þh i ¼ @c kð Þj j2D E

:

The gradient similarity index involving the gradients of two dif-

ferent Gaussian functions is obtained as:

rab� �2¼ abð Þ12

12aþ bð Þ

8>>>:9>>>;

5

; (10)

which is nothing else than the fifth power of the ratio between

the geometric and the arithmetic mean of the exponents of the

involved Gaussian functions.

2197Density Gradient Quantum Similarity


Of course, as the ratio between these means is less or equal

to the unit, it is always obtained that:

0 < rab� �2� 1;

so, if and only if both exponents are the same, one arrives to the

upper bound range of the similarity index: (raa)2 5 1.

Simple Overlap Similarity Integrals

It is indeed interesting to compare this result with the ones pro-

vided by simple overlap similarity measures, using the same

density GTO functions as before when gradients were analyzed.

Afterward, one must compute the overlap self-similarity in the

form of the Euclidean norm:

c að Þj j2D E

¼ N2 2að ÞZD

exp �4a rj j2� �

dr

¼ N2 2að Þ i0 4að Þð Þ3¼ ap

8: 9;32

with the formally equivalent result for the other GTO density:

c bð Þj j2D E

¼ bp

8>: 9>;32

:

Consequently, the corresponding overlap similarity measure

between both functions appears to be:

c að Þc bð Þh i ¼ N 2að ÞN 2bð ÞZD

exp �2 aþ bð Þ rj j2� �

dr

¼ N 2að ÞN 2bð Þ i0 2 aþ bð Þð Þð Þ3¼ 2abp aþ bð Þ

8>>: 9>>;32

¼ kp

8>: 9>;32

Thus, owing to this result, as it was done before when gradient

similarity was studied one can write:

c að Þc bð Þh i ¼ c kð Þj j2D E

:

Moreover, one can express the gradient similarity integrals as

obtained in eq. (8) in terms of the overlap similarity as:

@c að Þ@c bð Þh i ¼ 6k c að Þc bð Þh i ¼ 6k c kð Þj j2D E

:

In this case, one can write the overlap similarity index easily:

r0;ab� �2¼ abð Þ12

12aþ bð Þ

8>>>:9>>>;

3

Obtaining in this way the similarity index expression as the third

power of the ratio between geometric and arithmetic means of

the involved exponents.

Similarity Indices Analysis

Between the present and the former gradient indices, one will

consequently have the following relationship:

r0;ab� �2� rab

� �2thus, even in this simple case, the similarity indices between the

overlap similarity measures and the gradient integrals will be

somehow different as different are the similarity integrals.

To see this kind of relationship in a better way, suppose one

can write:

b ¼ x2a ^ x 2 R:

Then, the ratio of both involved means can be written as:

h xð Þ ¼ abð Þ1212aþ bð Þ ¼

2 xj j1þ x2ð Þ ; (11)

which is a function that can be easily visualized within any

power and that can be used in order to assess the variation of

both similarity indices. See Figure 1 for more details, where just

the function h(x) first power is represented.The function (11) shown earlier has a maximal value:

h(max)(1) 5 1, becoming null at zero and infinity. Besides, the

function h(x) can be used to measure the difference between two

GTO basis functions when employed in linear combinations in

order to obtain adequate atomic or molecular orbitals.

Even-Tempered Sequences of GTO

For instance, one can use the similarity index results obtained so

far within even- or well-tempered GTO sequences123–125 in order

to assess the adequate choice of exponent parameters. In fact,

even-tempered sequences of exponents {1k|k 5 1, p} rely on the

simple expression:

Figure 1. Graphical representation [eq. (11)] of the ratio between

geometric and arithmetic means of two exponents {a, b 5 x2a}.



1k ¼ abk�1

with the requirement: b [ R1 ^ b = 1. Thus, the similarity indi-

ces between two basis set orbitals can be expressed as a power

of the function h(x), which in this case can be written as:

h k; lð Þ ¼ 1k1lð Þ12121k þ 1lð Þ ¼

bkþl�2� �1

2

12bk�1 þ bl�1ð Þ (12)

Supposing that: l 2 k 5 d ^ d � 0 ? l 5 k 1 d, one arrives to

the expression:

h d; bð Þ ¼ 2bd2

1þ bd

Which it is essentially the same as eq. (12). Consequently, Fig-

ure 1 depicts how the similarity index can vary as x2 : bd

varies. It is interesting to note that different pairs of b \ 1 and

b [ 1 values can provide the same similarity index. As an

example, the value: hðxÞ ¼ 12can be obtained with x ¼ 2� ð3Þ12.

A Slightly Complex Example

An analogous analysis can be made as in the previous one cen-

ter treatment, but now with simple Gaussian functions having

centers {A,B}, considered separated at a certain squared dis-

tance: R2AB 5 |A 2 B|2, one from another. Then the functions to

be employed can be written as:

c r� Ajað Þ ¼ N 2að Þ exp �2a r� Aj j2� �

(13)

and

c r� B bjð Þ ¼ N 2bð Þ exp �2b r� Bj j2� �

(14)

possessing unit Minkowski norms as in the previous studied

cases. Their gradients are related to the gradients of formerly

studied Gaussian functions, so one can write:

@c r� A ajð Þ@r

¼ �4aN 2að Þ r� Að Þ exp �2a r� Aj j2� �

A similar expression holds for the other GTO function. To com-

pute the gradient similarity integral between both GTO func-

tions, one must use the well-known theorem about products of

GTO centered at different sites, as the dot product of gradients

becomes:

@c r� A��a� �

@r

�� @c r� B��b� �

@r

� �¼ 16ab

4abp2

8>: 9>;32

3 r� Að Þ; r� Bð Þh i exp �2a r� Aj j2� �

exp �2b r� Bj j2� �

ð15Þ

Then, as it is well known, the product of Gaussian exponentials

can be substituted by the following expression16,126,127:

exp �2a r� Aj j2� �


¼

exp �2ab A� Bj j2

aþ bð Þ

8>>>:9>>>; exp �2 aþ bð Þ r� Pj j2

� �

with the point P defined as: P ¼ aAþbBaþb , one can substitute this in

the former expression (15) and in order to obtain the final inte-

grand one shall compute the adequate expression for the scalar

product:

r� Að Þ; r� Bð Þh i ¼ r� Pþ P� Að Þ; r� Pþ P� Bð Þh i¼ r� Pj j2þ r� Pð Þ; 2P� Aþ Bð Þð Þh i þ P� Að Þ; P� Bð Þh i

Therefore, three different integrals have to be evaluated in this

case, that is:

a.

I0 ¼ZD

exp �2 aþ bð Þ r� Pj j2� �

dr

which using the previously defined integral i0(a), it is readily

obtained as:

I0 ¼ i0 2 aþ bð Þð Þð Þ3¼ p2 aþ bð Þ

8>>: 9>>;32

(16)

b.

I1 ¼ZD

ðr� PÞ�ð2P� ðAþ BÞÞ expð�2ðaþ bÞjr� Pj2Þdr

which it is related to the null integral:

i1 að Þ ¼Z þ1

�1x exp �ax2

� �dx ¼ 0; (17)

thus: I1 5 0

c. Finally, the following integral has to be found:

I2 ¼ZD

r� Pj j2exp �2 aþ bð Þ r� Pj j2� �

dr

which it is obviously associated to the integrals i0(a),i2(a),that is:

I2 ¼ 3i2 2 aþ bð Þð Þ i0 2 aþ bð Þð Þð Þ2¼ 3

2

p

8 aþ bð Þ38>>>:

9>>>;12 p

2 aþ bð Þ �

Thus, the gradient similarity integral sought in this way, can be

written in the present case as the integral of the result of the

scalar product between the gradients:



@c r�A ajð Þ@r

��@c r�B bjð Þ@r

� �¼

41

p

8>: 9>;32 2ab

aþ bð Þ8>>: 9>>;5

2 3

2� 2ab

aþ bð ÞR2AB

8>>: 9>>;exp � 2abaþ bð ÞR

2AB

8>>: 9>>;(18)

which one must expect that becomes coincident with the previ-

ous result when R2AB 5 0, as appears in eq. (8).

The aforementioned expression can be easily simplified using

the constant parameter k, depending of the involved GTO expo-

nents as defined in eq. (9). In this way eq. (18) can be simply

written as:

@c r�A ajð Þ@r

��@c r�B bjð Þ@r

� �¼4

k5

p3

8>>>:9>>>;

12 3

2�kR2

AB

8>: 9>;exp �kR2AB

� �:

Then, the corresponding gradient similarity index can be easily

computed as:

r2AB¼2 abð Þ12aþbð Þ

8>>>:9>>>;

5

1�2

3kR2

AB

8>: 9>;2

exp �2kR2AB

� �

Again the final expression becomes proportional to the fifth

power of the ratio between geometric and arithmetic means,

when the distance becomes null as in eq. (10).

The interesting thing here is to obtain the expression of the

similarity index when both functions bear the same exponent but

are at a non-null distance apart; that is:

a ¼ b ! k ¼ a ! r2AB ¼ 1� 2

3aR2

AB

8>: 9>;2

exp �2aR2AB

� �(19)

Just calling: c 5 2aR2AB, one can write:

r2AB ¼ 1� 1

3c

8>: 9>;2

exp �cð Þ

which is a convenient expression in order to visualize the form,

taken by the gradient similarity index, when both the distance

and exponent varies, see Figure 2, for more details.

In fact, when c 5 0 the similarity index becomes the unit,

but two extrema points are also present: a minimum at c 5 3,

where the index becomes null and a relative maximum at c 5 5,

where it takes the value: r2AB ¼ 49expð�5Þ.

Also, Figure 3 gives the detailed behavior of this similarity

index function in the appropriate ranges of both the function r2ABand the variable c, showing a hidden structured function behavior.

Overlap Similarity Measures

It could be also interesting to compare the previous density gra-

dient results with the simple overlap similarity measures, as it

has been previously done in the examples of the preceding

sections.

For that purpose one has to seek for the integral between the

Gaussian functions (13) and (14), that is:

c r� A ajð Þc r� B bjð Þh i ¼N 2að ÞN 2bð Þ

ZD

exp �2a r� Aj j2� �


dr

which can be evaluated as the integral between two spherical

Gaussian functions. In turn, this fact is a well-known result, see

for example ref. 128:

c r� A ajð Þc r� B bjð Þh i ¼

N 2að ÞN 2bð Þ p2 aþ bð Þ

8>>: 9>>;32

exp � 2abaþ b

R2AB

8>>: 9>>;¼ 1

p

8>: 9>;32 2ab

aþ bð Þ8>>: 9>>;3

2

exp � 2abaþ b

R2AB

8>>: 9>>; ð20Þ

Overlap self-similarity integrals are readily computed to be:

Figure 2. Representation of a simplified form of the gradient simi-

larity index.Figure 3. Detailed scaled range of the gradient similarity index.



c r� A ajð Þj j2D E

¼ ap

8: 9;32

;

therefore, the overlap similarity index becomes:

r20;AB ¼ 2 abð Þ12aþ b

8>>>:!3

exp � 4abaþ b

R2AB

8>>: 9>>;

which, when the distance becomes zero, transforms in the al-

ready computed similarity index depending on the third power

of the ratio of the geometric and arithmetic means of the expo-

nents. When both exponents are the same, the similarity index

yields a simple exponential function, which is coincident with

the exponential term of the gradient case:

r20;AB ¼ exp �2aR2AB

� �:

Thus, one can easily see that the similarity gradient index (19)

has a richer structure than the similarity overlap one.

Gradient Similarity in a Promolecular ASA

Background

The analysis performed beforehand can be immediately applied

to promolecular density functions constructed under ASA tech-

nique.112–119 In this approach, the molecular densities are con-

structed with a form for a molecular structure A like:

qA rð Þ ¼XI2A

QAI qI r� RA

I

� �;

where {QAI } are the set of nuclear charges or overlap popula-

tions in case that some kind of polarization of the atoms of Acan be taken into account, fulfilling:

PI2A QA

I ¼ N, being N the

number of electrons of molecule A and the set {qI} is a set of

ASA atomic basis density functions, centered at the atomic sites

{RAI } of A. The atomic ASA density functions are defined in

turn as:

qI r� RAI

� � ¼ Xl

wlcl r� RAI

��2al �

where {wl} and {al} are sets of coefficients and exponents opti-

mized for a specific atom I, given a fixed dimension of the sub-

space subtended by the spherical GTO {cl}, associated to unit

Minkowski norms: Vl: hcli 5 1. In this way these functions are

of the previous studied type, as in eqs. (13) or (14). Information

about a wide variety of ASA basis set density functions can be

found at the web site of ref. 129.

In this mathematical setup, one can consider that ASA func-

tions are just shape functions without loss of generality. That is:

the Minkowski norms of the ASA basis density functions being

unity, as the ASA basis functions employed here have this prop-

erty too, fulfilling:

qIh i ¼Xl

wl cl� ¼

Xl

wl ¼ 1:

The gradient of a promolecular density function will become

straightforwardly expressed as:

rqA rð Þ ¼XI2A

QAI rqI r� RA

I

� �:

The basis density function gradient can be formally written as:

rqI r� RAI

� � ¼ Xl

wlrcl r� RAI al��

:

Therefore, the integrals needed to compute gradient similarity

integrals between two molecular structures are the same as the

ones already computed, as shown in eq. (18). That is, it can be

written:

rqA rð Þ��rqB rð Þ

� �¼

XI2A

XJ2B

QAI Q

BJ rqI r� RA

I

� ��rqJ r� RBJ

� �� ð21Þ

and then:

rqI r� RAI

� ��rqJ r� RBJ

� �� ¼

Xl

Xm

wlwm rcl r� RAI al�� rcm r� RB

J

��am ��

The integral set: {h!cl(r 2 RAI |al)|!cm(r 2 RB

J |am)i} involving

the ASA basis functions can be evaluated using the result of the

eq. (18).

Hybrid Density–Density Gradient Similarity

There is another possibility which can be added to the gradient

similarity integrals and the overlap similarity itself: the feasible

connection of density function and its own gradient. Here will

be briefly discussed such a prospect.

The simplest integral involving a unique GTO and its gradi-

ent, is irrelevant from the computational point of view, as one

can write:

c að Þ@c að Þhj ii ¼ �4aN2 2að ÞZD

r exp �4a rj j2� �

dr ¼ 0j i



which is a three-dimensional null vector, owing to the fact that

the three integrals defining the resultant vector components

correspond to three integrals of the type:i1(a) as defined in

eq. (17). The same can be said when dealing with integrals of

different exponents, centered at the same site. That is, one can

also write: |hc(a)@c (b)ii 5 |0i from the same previously

adduced reason, when a 5 b.However, this is not so, for instance, when the gradient and

the compared GTO are located at different positions in space. In

this case one can write the following product as:

c r� A ajð Þ @c r� B bjð Þ@r

¼

� 4b4abp2

8>: 9>;32

r� Bð Þ exp �2a r� Aj j2� �


As a result, with similar manipulations as those employed when

dealing with the product of two gradients, using again the nullity

of the integrals with the first power of the electron position, the

integral I0 can be written, as defined in eq. (16). Then, a trivial

manipulation of the vector positions difference and the previous

definition of the parameter k in eq. (9) provides:

c r� A ajð Þ @c r� B bjð Þ@r

� ��

¼ 2 kkp

8>: 9>;32

exp �k A� Bj j2� �" #

B� Að Þ

An adequate procedure will consists to define the Euclidean

norm of the vector integral as defined earlier, obtaining in this

manner a positive definite expression. Then, employing the

squared distance between the function and gradient centers can

be written as:

hAB¼ c r�A��a� �@c r�B

��b� �@r

� �� c r�A��a� �@c r�B

��b� �@r

� �* +¼

4k5

p3

8>>>:9>>>;exp �2k A�Bj j2

� �" #A�Bj j2¼4R2

AB

3k5

p3

8>>>:9>>>;exp �2kR2

AB

� �" #

The resulting function has a shape which can be seen displayed in

the Figure 4, where the parameter k 5 1 has been employed. The

maximum location and height will depend on the parameter k.As such a similarity descriptor is by itself a Euclidian norm,

there is not easy to construct from such a norm other derived

similarity indices.

General GTO Gradient Similarity Integrals

As it is customary since long time ago, a general GTO can be

written as the product of a s-type GTO, like the ones which

have been studied in the previous sections plus an angular part,

which can be written as:

X n; r� Að Þ ¼Y3I¼1

xI � AIð ÞnI

where the three-dimensional vector A 5 {AI|I 5 1,3} is the cen-

ter of the GTO and the vector n 5 {nI|I 5 1,3} is an integer set

of three elements producing s-, p-, d-. . .type GTO when hni 50,1,2,. . ., respectively.

Thus, taking such preliminaries into account, a general GTO

can be written, using this conventional notation, as:

C n; r� A; að Þ ¼ N n; að ÞX n; r� Að Þ exp �a r� Aj j2� �

: (22)

Here the normalization factor can be computed in such a way

that the corresponding GTO Minkowski norm becomes unity:

C n;r�A;að Þj j2D E

¼N2 n;að Þ X 2n;r�Að Þexp �2a r�Aj j2� �D E

¼1

This aforementioned norm is related to the integral122:

I n;að Þ¼Z þ1

�1x2ne�2ax2dx¼ 2n�1ð Þ!!

22nþ1anp2a

8: 9;12

therefore:

X 2n;r�Að Þexp �2a r�Aj j2� �D E

¼Y3I¼1

2nI�1ð Þ!!22nIþ1anI

8>>>:9>>>; p

2a

8: 9;32

implying that the normalization factor should be computed as:

N2 n;að Þ¼Y3I¼1

22nIþ1anI

2nI�1ð Þ!!

8>>>:9>>>; 2a

p

8>: 9>;32

:

The gradient of the function (22) can be written as:

Figure 4. The function hAB of hybrid density-gradient similarity

varying with center distance RAB , computed for the parameter

defined in eq. (9), with a value k 5 1.



@

@rC n;r�A;að Þ¼N n;að Þ N n;r�A;að Þj iexp �a r�Aj j2

� �

where the angular gradient vector can be formally defined by

means of:

N n;r�A;að Þj i¼ d nJ>0ð ÞnJX n�eJ ;r�Að Þ�2aX nþeJ ;r�Að Þ J¼1;3jf g ð23Þ

where the set {eJ|J 5 1,3} is the three-dimensional canonical basis

set forming the rows or columns of I3 the (33 3) unit matrix.

Similarity measures between two density gradient vectors, as

initially defined in eq. (2) between two different quantum

objects {P, Q}, could be easily constructed as in eq. (3)

gPQ ¼ZD

@qP@r

;@qQ@r

� �dr¼

4Xl2P

Xm2P

Xk2Q

Xr2Q

DP;lmDQ;kr vP;l rð ÞvQ;k rð Þ @vP;m rð Þ@r

;@vQ;r rð Þ

@r

� ��

¼ 4Xl2P

Xm2P

Xk2Q

Xr2Q

DP;lmDQ;krAl 2P Ak 2Q Am 2P Ar 2Q

l k m r

� �

� 4Xl2P

Xm2P

Xk2Q

Xr2Q

DP;lmDQ;krAl Ak Am Ar

l k m r

� �

Consequently, in the most complex case one is facing a hybrid

integral involving two different GTO functions and two different

GTO gradients, centered at four different points of three-dimen-

sional space. Owing to the previous definitions it can be written:

Al Ak Am Ar

l k m r

� �¼Nðnl;alÞN nk;akð ÞN nv;avð ÞN nr;arð Þ

hX nl;r�Al� �

X nk;r�Akð ÞhN nm;r�Am;amð ÞjN nr;r�Ar;arð Þiexp �aljr�Alj2

� �exp �akjr�Akj2

� �3exp �amjr�Amj2

� �exp �arjr�Arj2

� �i

Where the symbol: hX(nm, r 2 Am, am)|X(nr, r 2 Ar, ar)i standsfor the scalar product of the elements of the vectors |X(n,r 2 A,

a)i constructed as previously defined in eq. (23).

To compute the implied four center integral, the four centers

can be transformed into a unique one in a similar way as the

repulsion integrals are manipulated.126,127,130 First it can be

made:

expð�aljr� Alj2Þ expð�akjr� Akj2Þ ¼

exp � alakjAl � Akj2ðal þ akÞ

8>>>:9>>>; expð�ðal þ akÞjr� Pj2Þ

with the new center defined as:

P ¼ alAl þ akAk

al þ ak� �

also:

P� Al ¼ alAl þ akAk � al þ ak� �

Al

al þ ak� � ¼ ak Ak � Al

� �al þ ak� �

and a similar form for the remaining product:

exp �am r� Amj j2� �

exp �ar r� Arj j2� �

¼

exp � amar Am � Arj j2am þ arð Þ

8>>>:9>>>; exp � am þ arð Þ r�Qj j2

� �

with the new center defined as:

Q ¼ avAv þ arAr

av þ arð Þ :

So the similarity integral can be rewritten as:

Al Ak Am Ar

l k m r

� �¼

Nðnl; alÞN nk; akð ÞN nv; avð ÞN nr; arð Þ

exp � alakjAl � Akj2

al þ ak8: �

8>>>>>>:9>>>>>>; exp � amarjAm � Arj2

am þ arð Þ

8>>>:9>>>;

hX nl; r� Al� �

X nk; r� Akð ÞhN nm; r� Am; amð ÞjN nr; r� Ar; arð Þiexp � al þ ak

� �jr� Pj2� �

exp � am þ arð Þjr�Qj2� �

i

and constructing a new composite center: S ¼ ðalþakÞPþðamþarÞQðalþakþamþarÞ

one arrives at the final form:

Al Ak Am Ar

l k m r

� �¼

Nðnl; alÞN nk; akð ÞN nv; avð ÞN nr; arð Þ

exp � alakjAl � Akj2

al þ ak8: �

8>>>>>>:9>>>>>>; exp � amarjAm � Arj2

am þ arð Þ

8>>>:9>>>;

exp � al þ ak� �

am þ arð ÞjP�Qj2al þ ak þ am þ ar� �8>>>:

9>>>;hX nl; r� Al

� �X nk; r� Akð ÞhN nm; r� Am; amð ÞjN nr; r� Ar; arð Þi

exp � al þ ak þ am þ ar� �jr� Sj2

� �i

In the expression shown earlier, the integral involves a product

of two GTO angular parts and another set of angular parts,

which correspond to the sum of the three elements of the scalar

product. These elements are associated to the resulting angular

parts of two GTO gradients. This setup yields, in the most compli-

cated case, six integral elements to be taken into account. Handling

the integration of the GTO angular parts is a well known solved

problem,126,127,130,131 due to this it will not be further developed.



Multiple Density Gradient Integrals

The definition of the density gradient integrals, as it has been

described in the preceding part of this study, can be generalized

in the same way as it has been previously studied.9 This leads in

a straightforward manner to multiple density gradient similarity

integrals. As the gradient density integrals are scalar products,

care must be taken to defining them in a coherent way with

respect to the associate norms in the fashion a recent study

points out.132 Triple9 and multiple similarity integrals15 may

play a leading role when quantum QSPR55 is developed beyond

a first order approach.47 Therefore, they are included here for

the sake of completeness.

Triple Density Gradient Integrals

The definition of a multiple density gradient integral can be eas-

ily written after a generalization of the current integral features

employed in the previous parts of this article. For example, one

can use the following conventional symbols, which constitute a

generalization of the scalar product concept:

gABC ¼ @qA@r

� @qB@r

� @qC@r

� �¼

ZD

@qA@r

;@qB@r

;@qC@r

� �dr

¼XI

ZD

@qA@xI

@qB@xI

@qC@xI

dr

Such a task as obvious as it seems, has to be contrarily studied

with care; for example, in the way as it has already been ana-

lyzed.132 The reason of this concern is simple. From the usual

scalar product involving two vectors, the Euclidian norm is eas-

ily deduced. This must be the case within multiple scalar prod-

ucts: at least for the sake of coherence, the associated general-

ized norms133,134 have to be defined as a particular case of such

vector operations.

On the other hand, Euclidian and generalized higher order

norms are positive definite structures. Otherwise, one should

only speak of pseudonormsy perhaps. When dealing with the tri-

ple scalar product, such scalar product-norm coherence shall be

already met. When studying the triple density overlap integrals,

see for example9:

zABC ¼ qAqBqCh i ¼ZD

qA rð ÞqB rð ÞqC rð Þdr 2 Rþ ! zAAA

¼ qAj j3D E

2 Rþ

such a problem will never appear, since density functions are

positive definite. However, the density gradient components are

nondefinite real functions, which can give real integral values,

as it has been previously pointed out. Therefore, one will have

in general:

gABC 2 R ! gAAA 2 R:

That is: the third order density gradient norm, defined as a par-

ticular case of the triple density gradient similarity integral,

where the three implied densities are the same, is a pseudonorm.

To force the positive definite nature of the third order norm, as

it can be obtained from triple scalar products and insuring its

positive definiteness, there is no other alternative than to get rid

of the potential negative signs in the range of one of the gra-

dients. For example, using the optional definition of defining the

triple scalar product as follows:

gcABC ¼ @qA@r

� @qB@r

� @qC@r

��

� �¼

ZD

@qA@r

;@qB@r

;@qC@r

��

� �dr

¼XI

ZD

@qA@xI

@qB@xI

@qC@xI

��dr

where one of the gradients has been employed with its elements

transformed into absolute values. However, such a procedure

will not be still adequate unless, in turn, every gradient could

appear with the same positive definite signature. To circumvent

this problem maybe an average shall be sought like:

gABC ¼ 1

3

@qA@r

�� @qB@r

� @qC@r

� �þ @qA

@r� @qB

@r

�� @qC@r

� �

þ @qA@r

� @qB@r

� @qC@r

��

� �!¼ 1

3gaABC þ gbABC þ gcABC

� �

so the triple density gradient norm could be computed as:

gAAA ¼ @qA@r

�� @qA@r

� @qA@r

� �¼

XI

ZD

@qA@xI

�� @qA

@xI

8>: 9>;2

dr 2 Rþ

which is a positive definite integral. The problem will appear

when taking into account the absolute value density gradient ele-

ments within the integrand, but this is a numerical problem which

can start to be solved redefining the absolute value function, in

terms of logical Kronecker deltas, see for example.135–137 For

instance:

8xI : @qA@xI

�� ¼ d

@qA@xI

> 0

8>: 9>;� d@qA@xI

< 0

8>: 9>;� @qA@xI

or using a reliable numerical lengthy integration.

Multiple Gradient Density Integrals

The scalar product-norm coherence problem studied earlier con-

cerning triple gradient integrals, will be present when dealing with

similarity integrals with odd number of density gradient products,

not when an even number of density gradient integrals are dealt

with. This is so, because the corresponding norm will be associ-

ated to positive definite even powers of the density gradient ele-

ments. However, averages of the integrals with an absolute value

yDefining as a pseudonorm some repeated scalar product of the same

vector, producing just a real number not necessarily positive definite.



of the gradient component can be described in the same way as in

the triple gradient integral case. This can be formally written as

follows. First define a vector containing indices which can be

attached to every density function, taking part in the integral of

orderM, that is: hL|5 (L1,L2,. . .LM), then two cases are present:

a.

M ¼ 2N ! g Lh j ¼X3I¼1

YMK¼1

@qLK rð Þ@xI

* +

b.

M ¼ 2N þ 1 ! g Lh j ¼ 1

M

X3I¼1

XMK¼1

@qLK rð Þ@xI

�� Y

J 6¼K

@qLJ rð Þ@xI

8>>>>:9>>>>;

* +

In this way, the multiple density gradient similarity integrals can

be generated coherently with the corresponding generalized

norms.

Some Numerical Examples

In this last section, some numerical examples will be provided.

The readers can assess in this way coincident and different fea-

tures of the density gradient similarity integrals, when compared

with the density overlap similarity measures. The present calcu-

lations have been performed under the so called ASA frame-

work. The basis sets chosen for the present calculations can be

found in the website129: and correspond to the fitting of 3-21G

and 6-311G basis sets atomic density results with specific linear

Table 1. Atomic Density Overlap and Density Gradient Self-Similarities

in Units of Z22 for the Chosen ASA Basis Sets.

Z ATOM

3–21G 6–311G

Overlap Gradient Overlap Gradient

1 H 0.0412 0.1595 0.0396 0.1565

2 He 0.1880 2.3297 0.1916 2.4728

3 Li 0.3444 10.3726 0.3509 11.1342

4 Be 0.5192 28.9938 0.5371 31.6229

5 B 0.6920 61.0553 0.7015 64.7845

6 C 0.8719 110.9789 0.8847 117.4829

7 N 1.0597 182.4697 1.0760 192.6864

8 O 1.2558 279.3036 1.2728 292.9638

9 F 1.4620 405.0913 1.4815 424.6262

10 Ne 1.6808 564.3417 1.7015 590.2282

11 Na 1.9024 759.5095 1.9334 804.7657

12 Mg 2.1454 1007.5625 2.1735 1065.9021

13 Al 2.3889 1301.6655 2.4204 1377.8610

14 Si 2.6366 1651.4605 2.6713 1744.8752

15 P 2.8900 2055.3427 2.9274 2172.0857

16 S 3.1478 2522.9314 3.1876 2662.4022

17 Cl 3.4101 3057.0333 3.4525 3227.4186

18 Ar 3.6769 3661.4306 3.7214 3867.8483

19 K 3.9417 4322.3298 3.9952 4588.0335

20 Ca 4.2191 5085.1347 4.2743 5399.1902

21 Sc 4.4915 5927.6999 4.5528 6325.1906

22 Ti 4.7653 6860.1060 4.8290 7320.3314

23 V 5.0410 7887.9377 5.1070 8414.1331

24 Cr 5.3103 8972.1183 5.3870 9613.9196

25 Mn 5.5992 10228.3665 5.6691 10921.5886

26 Fe 5.8811 11557.8625 5.9539 12324.5798

27 Co 6.1650 12997.1184 6.2409 13884.4084

28 Ni 6.4527 14555.2824 6.5302 15547.7904

29 Cu 6.7920 16230.0767 6.8218 17328.7647

30 Zn 7.0343 18024.4120 7.4711 19552.4502

31 Ga 7.3381 19974.3424 7.4142 21226.4605

32 Ge 7.6456 22051.8083 7.7207 23436.7388

33 As 8.0132 24194.9131 8.0287 25786.3090

34 Se 8.2715 26587.7116 8.3397 28293.5523

35 Br 8.6051 29080.6330 8.6513 30962.2816

36 Kr 8.9072 31732.3444 8.9714 33779.0080

Table 2. Self-Similarity of Density Overlap and Gradient Density for

Some Assorted Molecules, Scaled by the Square of the Number of

Electrons in the Molecular Structure.

Molecule Overlap Gradient

Borine carbonyl 0.2730 49.3216

Methyl cyanide 0.2545 37.0894

Methyl isocyanide 0.2546 37.2458

Diazomethane 0.2918 46.5150

Ketene 0.3144 56.5872

Cyanamide 0.3069 49.6907

Table 3. Density Overlap Similarity Measures and Gradient Density

Similarity Integrals for Some Assorted Molecules.

Molecule Overlap Gradient

Borine carbonyl

Diazomethane 0.1825, Dz 5 0.00 16.9555, Dz 5 0.00

Methyl cyanide 0.2434, Dz 5 0.15 36.6173, Dz 5 0.15

Methyl isocyanide 0.2154, Dz 5 0.20 28.0477, Dz 5 0.20

Ketene 0.1488, Dz 5 0.00 15.3020, Dz 5 2.00

Cyanamide 0.2110, Dz 5 0.05 31.7129, Dz 5 0.05

Diazomethane

Methyl cyanide 0.2096, Dz 5 0.20 27.7636, Dz 5 0.25


Ketene 0.2422, Dz 5 0.00 28.6160, Dz 5 0.00

Cyanamide 0.2676, Dz 5 0.10 34.1580, Dz 5 0.10

Methyl cyanide


Ketene 0.1194, Dz 5 0.00 18.2584, Dz 5 4.65

Cyanamide 0.2118, Dz 5 0.00 18.5480, Dz 5 0.00

Methyl isocyanide

Ketene 0.1470, Dz 5 1.85 25.8758, Dz 5 1.85

Cyanamide 0.1951, Dz 5 0.00 19.1350, Dz 5 0.00

Ketene

Cyanamide 0.2962, Dz 5 0.20 46.7772, Dz 5 0.20

The Dz next to the similarity results correspond to a z axis translation

in a. u. of the second molecule with respect to the first (displaced to the

left of the column) providing a maximal similarity. Results are scaled by

the product of the number of electrons of both molecular structures.



Figure 5.

Figure 6.

combinations of 1s GTO functions. Both atomic and molecular

calculations have been performed with two Fortran 90 programs,

which can be downloaded from the web site of ref. 138.

Atomic Self-Similarities

Atomic overlap self-similarities have been published in earlier

work, when developing the ASA approximation.112–119 Here, the

results for both basis sets are also presented for both overlap and

gradient self-similarities and are resumed in Table 1. Self-similar-

ity results are written in units of Z22 in order to keep part of the

obtained values as they have been presented in the previously

quoted publications. There is another ASA basis set available from

H to Rn,115 based on an extensive Huzinaga basis set study,139,140

which will not be shown here because the results appear to be sim-

ilar to the ones in the Table 1. Details of this ASA Huzinaga fitting

can be also obtained in the website of ref. 129.

As expected, the numerical values obtained in both basis set

cases are quite similar. One must note the difference between

both overlap and gradient results, which as the atomic number

increases gradients become several orders of magnitude bigger

than the corresponding overlap measures.

The approximate functions connecting similarities for both

basis sets appear to be the same, almost without change in the

correlation coefficient (R2 5 0.998) and the kind of power func-

tion exponent (gradSS � 170 ovlSS2.37).

Molecular Self-Similarity

All the corresponding self-similarities or pair similarity values,

which will be presented in Table 2 below, have been scaled by the

squared number of electrons of the involved molecular structures.

The reported calculations on the molecules in the tables and

figures of this work have been obtained from ref. 141, where the

molecular coordinates and Mulliken gross atomic populations

have been picked up to be employed in the ASA promolecular

density function construction and then in the computation of both

the density overlap measures and the density gradient integrals.

This has been done in order that the presented results can be

easily reproducible and use of standard quantum chemical pro-

grams will be not needed in this way here.

Molecular Similarity and Molecular Superposition

Using the molecules of the previous section, with the parameters

of the mentioned source,141 in this section the variation of the

density overlap and gradient similarity will be studied for molec-

ular pairs in terms of the relative position of one structure with

respect of the other. Table 3 provides information on the density

overlap and density gradient between each pair of elements of

the chosen molecular set.

It is interesting to note here, that in some cases the similarity

maximum with respect to the relative position between two mol-

ecules of the chosen set can differ from overlap to gradient simi-

larity integrals.

Visualization of the Overlap and Gradient Similarity

Some assorted examples have been selected here in order to vis-

ualize the self-similarity variation and bimolecular similarity,

when one of the involved molecular structures is moved over

the other. As will be seen from the following graphs, the shapes

of overlap and gradient appear similar until some parts of the

gradient variation are magnified, then in these locations it is

encountered a similar behavior as in the simpler cases, like the

ones depicted in Figure 3 studied before.

Though here, in these particular sections of the gradient den-

sity similarity graphs the integral becomes negative and shapes

Figure 5. (a) Density overlap of two borine carbonyl molecules superposition. The z axis correspondsto the translation of one of the two structures along the z coordinate. (b) Density gradient integral of

two borine carbonyl molecules superposition. The z axis corresponds to the translation of one of the

two structures along the z coordinate. The two flat minimal basins are enlarged in Figures 5c and 5d.

(c) Density gradient integral of two borine carbonyl molecules superposition. The z axis corresponds tothe translation of one of the two structures along the z coordinate. This corresponds to the enlargement

to the first minimal basin in Figure 5b. (d) Density gradient integral of two borine carbonyl molecules

superposition. The z axis corresponds to the translation of one of the two structures along the z coordi-nate. This corresponds to the enlargement to the second minimal basin in Figure 5b.

Figure 6. (a) Density overlap measure of borine carbonyl and methyl cyanide superposition. The z axiscorresponds to the translation of methyl cyanide along the z coordinate. (b) Density gradient integral of

borine carbonyl and methyl cyanide superposition. The z axis corresponds to the translation of methyl cy-

anide along the z coordinate. The two flat minimal basins are enlarged in Figures 6c and 6d. (c) Density

gradient integral of borine carbonyl and methyl cyanide molecules superposition. The z axis correspondsto the translation of methyl cyanide along the z coordinate. This corresponds to the scale enlargement to

the first minimal basin in Figure 6b. (d) Density gradient integral of borine carbonyl and methyl cyanide

molecules superposition. The z axis corresponds to the translation of methyl cyanide along the z coordi-nate. This corresponds to the scale enlargement to the second minimal basin in Figure 6b.



of two minima and a maximum appear, adapting quite accu-

rately to a quartic polynomial behavior.

In all the studied cases, there are found two regions presenting

such a characteristic behavior. One corresponds to a small dis-

placement of the z coordinate, while the other to a larger one.

Both molecular relative positions correspond to a superposition,

where two atoms of the involved molecular structures, are

slightly separated or coincident. These regions in the density

overlap variation correspond to large minima depressions. Fig-

ures 5 and 6 below show this behavior. To have a similar moving

pattern in all cases, only translations on the positive z axis havebeen considered.

The molecules in the graphs 5, 6, and 7 have been chosen

among a series of computations performed on the molecules of

the aforementioned Table 2, where their nature and some self-

similarity information is given. An atomic GTO basis set of

type 3-21G within promolecular ASA density functions has

been employed in all the computations. Translations have been

made from both molecular positions as the original coordi-

nates.

The Figure 7 set is chosen as an example, because it presents

a maximal feature out of the translation origin, and so the graphs

correspond to a slightly different behavior than the previous

ones.

As a general result one can see that the variation of both

overlap and gradient molecular similarities appear in the bulk

graphic representations as having not extremely different appear-

ances. The similarities are distinct in precise sections of the mo-

lecular superposition as the result that gradient similarity inte-

grals may become slightly negative.

Atom–Atom Superposition

The shapes of density overlap and gradient, as studied before,

suggest that among atoms some similar behavior could be

traced, possessing part of the molecular behavior. To visualize

Figure 7. (a) Density overlap measure of diazomethane and ketene superposition. The z axis correspondsto the translation of ketene along the z coordinate. (b) Density gradient integral of diazomethane and ke-

tene superposition. The z axis corresponds to the translation of ketene along the coordinate z. The two flat

minimal basins are enlarged in Figures 7c and 7d. (c) Density gradient integral of molecules diazomethane

and ketene superposition. The z axis corresponds to the translation of ketene along the z coordinate. Thiscorresponds to the scale enlargement to the first minimal basin in Figure 7b. (d) Density gradient integral

of diazomethane and ketene molecules superposition. The z axis corresponds to the translation of

ketene along the z coordinate. This corresponds to the scale enlargement to the second minimal basin in

Figure 7b.



this possible atom–atom similarity changes, several calcula-

tions involving the same atom and two different atomic ele-

ments have been performed in this section. The Figures 8 and

9 sets below show the atom–atom behavior for density over-

lap and gradient density similarities of some atom–atom

(C��C and N-P) similarity behavior. The blow-up of the tail

of the gradient density shows a comparable trend to the one

found in the molecular interaction for density gradients.

Therefore, the quartic features observed in molecular density

gradient graphs, within the regions where nearby atom-atom

superposition occurs, can be attributed to occurrences of this

sort already appearing in single atomic pair superposition

under ASA framework.

Such a density gradient characteristic features, appears suffi-

ciently interesting as to be studied in deep elsewhere. It can be

Figure 8. (a) C��C atomic density overlap similarity as a function

of the distance between both atoms. (b) C��C gradient density simi-

larity as a function of the distance between both atoms. (c) C��C

atomic density gradient similarity as a function of the distance

between both atoms. This corresponds to the tail blow-up of the Fig-

ure 8b.

Figure 9. (a) N-P atomic density overlap similarity as a function of

the distance between both atoms. (b) N-P atomic gradient density

similarity as a function of the distance between both atoms. (c) N-P

atomic density gradient similarity as a function of the distance

between both atoms. This corresponds to the tail blow-up of the Fig-

ure 9b.



potentially employed to circumscribe or signal the neighborhood

regions of the molecule–molecule superposition, lying next to

maximum density gradient similarity integrals. Further research

will be directed into this path.

Conclusions

Density gradient similarity integrals have been analyzed and

compared with the well-known density overlap similarity meas-

ures. Density gradient integrals, excepting a scale factor of about

two orders of magnitude, behave almost the same as overlap

density measures when observed without detail. Although both

similarity integrals appear in bulk features with quite an equal

behavior, the potential of the new similarity integrals lies on the

quite different particular features of the gradient similarity

unveiled along the present study, when compared with density

overlap.

The non positive definiteness of the density gradients makes

them essentially different from overlap in certain regions of the

similarity superposition space. This feature can be certainly

employed in the development of new superposition algorithms

of two molecular structures.

Density gradient similarity integrals can be considered as a

new family of quantum mechanical tools to assess similarity

between quantum objects.

Acknowledgments

L. D. Mercado work is associated to a research fellowship

attached to this project. Referee suggestions have permitted an

important improvement of this work.

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Commentaries on quantum similarity (1): Density gradient quantum similarity