Through-space and through-bridge interactions in the correlation analysis of chemical bonds

Computational and Theoretical Chemistry 1026 (2013) 72–77

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Through-space and through-bridge interactions in the correlation analysis of chemical bonds Dariusz W. Szczepanik ⇑, Janusz Mrozek Department of Computational Methods in Chemistry, Jagiellonian University, Ingerdena 3, 30-060 Cracow, Poland

a r t i c l e

i n f o

Article history: Received 19 September 2013 Received in revised form 14 October 2013 Accepted 15 October 2013 Available online 24 October 2013 Keywords: Orbital communication Through-space Through bridge Electron population Bonds correlation Joint-probability

a b s t r a c t The formalism of the through-space and through-bridge communications proposed by Nalewajski within the framework of the Orbital Communication Theory is used to evaluate the correlation effect between electron populations of two chemical bonds. Proposed in this paper purely-probabilistic formalism defines the correlation between two chemical bonds as determined by the difference between appropriately normalized fourth-order joint probabilities corresponding to direct (through-space) and indirect (through-bridge) communication within a four-state time-homogeneous Markov chain. Alternative bond-correlation coefficient introduced earlier by Yamasaki and Goddard is based on the concept of the hierarchy of statistical covariance operators. The proposed correlation coefficient requires computing only the second-order probability terms when working within the condensed-atomic resolution and thus they are far less computationally expensive making them suitable for correlation analysis of chemical interactions between large molecular fragments. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Within the framework of the Orbital Communication Theory (OCT) originally formulated by Nalewajski [1–15] a molecule is interpreted as a communication system consisting of: ‘‘source’’, ‘‘communication channel’’ and ‘‘output’’. The source is defined through a set of orthogonal atomic orbitals (AO) and the corresponding electron probabilities while the ‘‘output’’ of a molecular communication channel is defined through (not necessarily the same) a set of AOs with the corresponding electron probabilities. Molecular communication channel is uniquely determined by the conditional probability matrix in atomic orbital resolution, which elements stand for the probability of finding on particular atomic orbital an electron originated from another atomic orbital. The information flow through such orbital communication system is attributed to changes in electron occupations of particular atomic orbitals in the molecule. Information is dissipated in the communication channel, and the noise in the OCT is related to delocalization of electrons. The conditional entropy measures the amount of such scattering and is treated as a IT-covalency index of molecular system. A complementary descriptor called mutual information measures the amount of information which passes through the communication channel, thus it specifies to what a degree electrons are attributed to particular atoms; it usually reflects the IT-ionicity of the molecular system. ⇑ Corresponding author. Tel.: +48 12 663 22 13. E-mail address: [email protected] (D.W. Szczepanik). 2210-271X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2013.10.015

Seemingly, the correspondence between orbital interaction and orbital ‘‘communication’’ provided by the OCT comes across as somewhat weird and non-intuitive. However, it has recently been shown in [16] that such orbital-communication approach is entwined with the theory of Markov chains [17] and arises imperceptibly as a consequence of localizing of electron atomic populations within the density matrix of any mixed quantum–mechanical state. Moreover, adopting the orbital communication formalism as well as other tools based on the Information Theory [18–20] in description of the molecular electronic structures has turned out to be very useful in making semantically more precise such ambiguous chemical concepts as atom-in-molecule (AIM), bond multiplicity, bond ionicity, molecular similarity, molecular fragment, etc. In the present work we will take advantage of the recently proposed OCT-based concept of through-space and through-bridge communications proposed by Nalewajski [21–26] to probe the effect of correlation between electron populations of two chemical bonds. 2. Conditional probability Let us consider the one-determinant N-electron wave function of the ground-state molecular system, jU0i, which gives rise to following one-electron density,

qðrÞ ¼ N

XX l

m

Dl;m vyl ðrÞvm ðrÞ;

ð1Þ

73

D.W. Szczepanik, J. Mrozek / Computational and Theoretical Chemistry 1026 (2013) 72–77

where Dl,m stands for an element of the corresponding density matrix within representation of pure-state basis functions jvi and vl(r)  hrjvli. For one-determinant state functions of closed-shell systems within the representation of orthonormal basis functions the density matrix D is idempotent, which means that

D2 ¼ D;

X jDl;m j2 ¼ Dl;l :

or

ð2Þ

m

If jvi represents a set of atom-centered localized orthonormal atomic orbitals (e.g. [27–34]), the relation (2) is crucial for performing the electron population analysis within the resolution of condensed atoms {Xa}. Indeed, in such a case, a diagonal element of symmetric matrix D can be regarded as the effective electron probability in AO representation, Dl,l  pl, and hence the first-order electron probability in reduced atomic resolution reads

( PP

ð1Þ

¼

) Xa X Pa ¼ Dl;l ;

ð3Þ

l

with the appropriate normalization condition Xa X XX X Pa ¼ pl ¼ Dl;l ¼ 1:

a

a

l

ð4Þ

l

The second equality in (2) and the above equation give rise to the following symmetric matrix of second-order (joint) electron probabilities in the condensed AIM resolution,

( Pð2Þ ¼

Pa;b ¼

) Xb Xa X X jDl;m j2 ; l

m

satisfying

ð6Þ

a

b

As stated by the Bayes theorem [35], both electron probabilities of AIMs, (3) and (5), are directly related through the corresponding conditional probability,

Pa;b ¼ Pa Pbja ¼ Pb Pajb ¼ Pb;a :

ð7Þ

Therefore, for one-determinant closed-shell systems the conditional probability matrix within the resolution of atoms in molecule is the following function of density-matrix elements:

( P

ðcÞ

¼

Pbja

) Xb Xa X X 2 ¼ Pa jDl;m j : 1

l

ð8Þ

m

It has recently been argued elsewhere [16] that the above definition can be reformulated in terms of natural orbitals (NO) [36] and their occupation numbers to comprise also multi-determinant correlated wave functions (of both, closed- as well as open-shell electronic structures):

( P

ðcÞ

¼

1

Pbja ¼ Pa

Xb n Xa X X X

l

m i¼1

Xa ! Xc1 !    ! Xcr ! Xb ;

ðrþ1Þ

Pbja

¼ Pc1 ja    Pbjcr > 0;

ð10Þ

identifies an indirect communication, i.e. the rth order throughbridge atomic interaction. In accordance with [16], any electron probability distribution determined by the diagonal elements of idempotent density matrix D is stationary in the sense of Markov, i.e.,

P ¼ PPðcÞ :

ð11Þ

Moreover, as follows from the Perron–Frobenius theorem [37], the uniqueness of the stationary distribution P is always assured if

h ix lim PðcÞ ¼ 1P;

ð12Þ

x!1

where 1 stands for the column vector with all entries equal 1. Stationarity and uniqueness of the electron distribution (3) are indispensable conditions for performing the density-matrix based population analyses. It has been demonstrated in [16] that using both definitions, (8) and (9), allows one to straightforwardly decompose the overall population of electrons in a molecule into 1–, 2–, and up to (r + 1)-center atomic indices (atomic populations, bond orders, etc.). 3. The fourth-order joint probability

ð5Þ

XX X Pa;b ¼ Pa ¼ 1: a

(through-space atomic interaction) Xa ? Xb, while the cascade of (r + 1) serially-connected communications determined by condiðrþ1Þ tional probability Pbja ,

" ki;i Cl;i Cm;i

i1 X Cl;i Cm;i þ 2 Cl;j Cm;j

!#) :

j¼1

Now let us focus on the fourth-order joint electron probability within the resolution of condensed atoms in molecule. We will use as an example the molecular subsystem of four interacting atoms labelled A, B, C and D, assuming atoms in pairs (A, B) and (C, D) to be covalently bonded (all remaining interactions are taken to be non-covalent in character). In the approximation of non-correlated electron pairs of chemical bonds the average joint probability of simultaneous assignment of one electron to four different atoms can be expressed as the following arithmetic mean:

PA;B;C;D ¼

1 ðPA;B PC;D þ PA;C PB;D þ PA;D PB;C Þ: 3

ð13Þ

In order to include the relative correlation of all bonding electron pairs within a subsystem of four interacting atoms let us first reformulate the product of two second-order probabilities, e.g. PA,BPC,D, in accordance with Bayes theorem (7), i.e. as mean value of the following equivalent terms: PA;B PC;D ¼

 1 P BjA P A  PC P DjC þ P BjA PA  PD P CjD þ P AjB P B  PC PDjC þ PAjB PB  P D PCjD : 4 ð14Þ

The above equality clearly indicates that the reason for the limitations of the definition (13) lies in the exclusion of the correlation between electron populations on particular atoms of two different chemical bonds. For example, in the probability product PA,BPC,D interactions A  C, A  D, B  C and B  D are neglected, since

ð9Þ

PA  PC – PA;C ;

PA  P D – PA;D ;

  Here Cl;i are the elements of LCAO NO matrix C and {ki,i} stand for the elements of diagonal occupation matrix defined as k ¼ Cy DC. One should realize that if there is no interaction between atoms Xa and Xb then Pbja  Pb and Pajb  Pa and hence, the corresponding second-order (joint) electron probability can be straightforwardly factorized, i.e. Pa,b = PaPb. In accordance with the Orbital Communication Theory and the theory of time-homogeneous Markov chains, Pbja represents the probability of ‘‘observing’’ on atom Xb the electron originally assigned to atom Xa. Thus, Pbja > 0 gives rise to direct communication

PB  PC – PB;C ;

PB  P D – PB;D :

ð15Þ

Therefore, replacing all non-correlated first-order probability products in (14) by the corresponding second-order joint probabilities from (15) allows one to include the average influence of remaining subsystem interactions on A–B and C–D. Then, the desired fourthorder joint probability can be expressed as

PA;B;C;D ¼ where

1 ðPA;B:C;D þ PA;C:B;D þ PA;D:C;B Þ; 3

ð16Þ

74

PA;B:C;D ¼

PA;C:B;D ¼

PA;D:B;C ¼

D.W. Szczepanik, J. Mrozek / Computational and Theoretical Chemistry 1026 (2013) 72–77

 1 P BjA PA;C PDjC þ PBjA PA;D PCjD þ PAjB PB;C P DjC þ PAjB P B;D PCjD ; 4 ð17aÞ  1 P CjA PA;B PDjB þ PCjA PA;D PBjD þ PAjC PC;B P DjB þ PAjC P C;D PBjD ; 4 ð17bÞ  1 P DjA PA;C PBjC þ PDjA PA;B PCjB þ PAjD PD;C PBjC þ PAjD P D;B PCjB : 4 ð17cÞ

It has to be noticed, that formally there are 4! = 24 permutations for the four-state Markov chain. However, the joint probability (16) involves only 12 different terms due to the symmetry of Markov chains. Indeed, for exemplary communications ð0Þ

A ! B ! C ! D;

ð0Þ

D ! C ! B ! A;

Q A;B;C;D ¼ PA PBjA PCjB PDjC : Q D;C;B;A ¼ PD PCjD PBjC PAjB :

we can prove straightforwardly using Bayes theorem (7) that ð0Þ

Taking advantage of (10) allows one to recognize PA,B:C,D as the average joint probability of indirect interaction between two atoms involving the ‘‘bridge’’ of another two atoms, 

ð0Þ

PA;B:C;D  Q 4 ¼

 1  ð0Þ ð0Þ ð0Þ ð0Þ Q B;A;C;D þ Q B;A;D;C þ Q A;B;C;D þ Q A;B;D;C : 4

ð0Þ

½B ! A ! ½C ! D;

ð19aÞ

ð0Þ

½B ! A ! ½D ! C;

ð19bÞ

ð0Þ

½A ! B ! ½C ! D;

ð19cÞ

ð0Þ

½A ! B ! ½D ! C:

ð19dÞ

Q B;A;D;C ¼ PB PAjB PDjA PCjD : Q A;B;C;D ¼ PA PBjA PCjB PDjC : Q A;B;D;C ¼ PA PBjA PDjB PCjD :

ð0Þ

¼ PAjB PBjC PC;D ¼ PAjB P BjC PCjD PD ¼ Q D;C;B;A :

ð22Þ

ð18Þ

In the above definition each Q-term in brackets represents the weighted transition probability of the corresponding four-state Markov chains:

Q B;A;C;D ¼ PB PAjB PCjA PDjC :

Q A;B;C;D ¼ PA PBjA P CjB PDjC ¼ PA;B PCjB PDjC ¼ PAjB PB;C PDjC

Similarly, the sum of joint probabilities (17b) and (17c) can be regarded as the average fourth-order joint probability    1  ð1Þ ð1;2Þ ð1Þ ð2Þ ð1Þ P A;C:B;D þ P A;D:B;C  Q 4 ¼ Q 4 þ Q 4 ¼ Q C;A;D;B þ Q A;C;B;D 4  ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ þQ D;A;C;B þ Q A;D;B;C ;þQ C;A;B;D þ Q A;C;D;B þ Q D;A;B;C þ Q A;D;C;B ;

4. Correlation coefficient of chemical bonds In the present study we propose to take an advantage of joint   ð0Þ ð2Þ probabilities Q 4 and Q 4 in the quantitative description of the correlation effect between electron populations of two chemical bonds, A–B and C–D. The correlation coefficient of two chemical bonds, #A–BjC–D, within such a purely probabilistic approach is defined as the appropriately normalized difference between the probability of the fourth-order atomic communication involving  ð2Þ one covalent bond forming a diatomic bridge, Q 4 , and the corresponding probability representing communication cascade involv ð0Þ ing non-covalently interacting bridge-atoms, Q 4 :

   ð2Þ ð0Þ #A—BjC—D ¼ b  rA—BjC—D ¼ b Q 4  Q 4 ;

ð23Þ

where the normalization constant,

b¼



rA—BjA—B  rC—DjC—D

1=2

;

ð24Þ

ð20Þ

of the following four-state Markov chains: ð1Þ

A ! ½C ! ½B ! D;

ð21aÞ

ð1Þ

C ! ½A ! ½D ! B;

ð21bÞ

ð1Þ

A ! ½D ! ½B ! C;

ð21cÞ

ð1Þ

D ! ½A ! ½C ! B;

ð21dÞ

ð2Þ

A ! ½C ! D ! B;

ð21eÞ

ð2Þ

A ! ½D ! C ! B;

ð21fÞ

ð2Þ

C ! ½A ! B ! D;

ð21gÞ

ð2Þ

D ! ½A ! B ! C:

ð21hÞ

Q A;C;B;D ¼ PA PCjA PBjC PDjB : Q C;A;D;B ¼ PC PAjC PDjA PBjD : Q A;D;B;C ¼ PA PDjA PBjD PCjB : Q D;A;C;B ¼ PD P AjD PCjA PBjC : Q A;C;D;B ¼ PA PCjA PDjC PBjD : Q A;D;C;B ¼ PA PDjA PCjD PBjC : Q C;A;B;D ¼ PC PAjC PBjA PDjB : Q D;A;B;C ¼ PD P AjD PBjA PCjB :

In Eqs. (18), (19d), (20) and (21) superscript of each Q-term refers to the number of atoms originated from chemical bond C–D (A–B) and forming a bridge in indirect interaction between atoms A(C) and B(D) while subscript 4 stands for the overall number of interacting atoms in the subsystem in question. Each arrow in Markov chains represents a direct communication (conditioned interaction) between particular two atoms and square brackets are used to group one- and two-atomic bridges separating covalently bonded atoms.

assures the correlation coefficient #A–BjC–D to be in the range of 1 to 1. In brief, descriptor (23) provides the opportunity to recognize the character and evaluate the magnitude of the correlation between electron populations of two chemical bonds, A–B and C–D. In particular, #A–BjC–D > 0 gives rise to positive bond correlation which means that forming (breaking) the chemical bond A–B automatically implicates a simultaneous forming (breaking) the chemical bond C–D, and vice versa. Otherwise, i.e. in the case of negative bond correlation (#A–BjC–D < 0), forming (breaking) the chemical bond A–B causes a concurrent breaking (forming, enhancing) of the chemical bond C–D. In accordance to expectations, #A–BjC–D depends on bonding structure, the distance between constituent atoms as well as the nature of chemical bonds (covalent, non-covalent, ionic, etc.). As will be demonstrated in the next section, in the case of chemical bonds of type A–B–C order of magnitude of the quantity #A–BjB–C usually varies from ±101 to ±102, while in the case of bonds A–B and C–D separated by a large distance #A–BjC–D generally assumes values in the range of ±105–±102.

5. Results and conclusions In this section we examine to what extent the proposed correlation coefficients involving the through-space/through-bridge concept of chemical bonding deal with the description of molecular electronic structures and the prediction of reactivity trends in chemical reactions. All calculations have been performed using the standard quantum-chemistry software GAMESS [38,39] and the special program written by authors.

D.W. Szczepanik, J. Mrozek / Computational and Theoretical Chemistry 1026 (2013) 72–77

75

5.1. Correlation analysis of chemical bonds in substitution reactions In the first instance let us consider the following simple reaction of substitution of a hydrogen atom by a deuterium atom in the molecule H2:

H—H0 þ D ! ½H    H0    D ! H0 —D þ H:

ð25Þ 0

0

Correlation coefficients (23) of bonds H–H and H –D, Wiberg’s bond orders [40] as well as total energy changes related to the energy of the reaction intermediate were calculated for the reaction path depicted in Fig. 1 at the DFT/B3LYP [41,42] theory level, involving basis set aug-cc-pVTZ [43] and the representation of natural atomic orbitals (NAO) [28]. For transparency, the values of #H—H0 jH—D and DE (in [kcal mol1]) depicted in Fig. 1 have been multiplied by scaling factors of a# = 30 and aDE = 0.1, respectively. Approximated bond lengths in the intermediate state of the reaction (25) are: RHH0  RH0 D  0:9 Å. Thus, it follows from the analysis of bond orders that at the distance between H0 and D of about 2.3 Å no significant interaction can be observed. Admittedly, at this distance the correlation coefficient (23) assumes small positive value but due to W H0 —D  0 one should regard such correlation effect as being only of minor importance. At the distance of about 1.7 Å we can observe the appearance of destabilizing interaction between atoms which is reflected by a negative correlation of chemical bonds H–H0 and H0 –D (#H—H0 jH—D < 0). As follows directly from Fig. 1, correlation coefficient achieves the minimum at the reaction coordinate referring exactly to the activated complex [H  H0   D]. This fact allows one to identify without a second thought the interaction of electrons from populations of chemical bonds H–H0 and H0 –D as destabilizing the intermediate molecular structure and determining the direction in which the reaction (25) proceeds. Since there are only two negatively correlated bonds in the activated complex of the reaction (25), the problem of the prediction of products is somewhat trivial. However, in the case of larger polyatomics there is diversity of prospective reaction paths giving rise to different products. For instance, let us consider the following substitution reaction:

TiCl2 Hþ þ D—D0 ! ½TiCl2 HDþ2  ! TiCl2 Dþ þ H—D0 ;

Fig. 2. The activated complex of the reaction (26) with the values of Wiberg bond orders alongside the corresponding bonds and bond-correlation coefficients of selected interactions. Method: DFT/B3LYP/aug-cc-pVTZ/NAO.

coefficient (23) clearly indicates that interaction of atoms Ti and D significantly impairs the chemical bond D–D0 . Consequently, the chemical bond H–D0 is being formed and, since H–D0 is negatively correlated with Ti–H, the latter is being broken as the reaction proceeds. Such a scenario is in full agreement with (26). Now let us take a closer look at relatively weak atomic interaction Ti  D0 . Its strong positive correlation with bonds Ti–D and Ti–H seems to be somewhat obvious since Ti  D0 is being formed when the D2 molecule approaches cation TiCl2H+ and is being broken when HD0 distances from the product TiCl2D+. However, the mutual antagonism of both these actions gives rise to stabilization of Ti  D0 bond in the activated complex TiCl2HD2+. Indeed, the corresponding values of (23) clearly indicate that this relatively weak atomic interaction effectively reconciles two negatively correlated bonds, i.e. D–D0 and H–D0 . This effect leads to slight but noticeable distortion of the picture of atomic interactions in the intermediate state of reaction (26) since, contrarily to expectations, the resulting coefficient of the ‘‘screened off’’ correlation between electron populations of bonds H–D0 and Ti–D (as well as D0 –D and Ti–H) assumes negative value.

ð26Þ

and try to predict the products basing only on the analysis of bond orders and correlation coefficients of the activated complex TiCl2 HDþ 2 , calculated using DFT/B3LYP/aug-cc-pVTZ/NAO method and presented in Fig. 2. As follows from analysis of Wiberg indices the strongest bonds in the molecule are: Ti–H, Ti–D, D–D0 and H–D. Correlation

Fig. 1. The values of relative total energy DE in (kcal mol1), Wiberg bond orders W H—H0 and W H0 —D as well as the bond-correlation coefficient #H—H0 jH0 —D on the path of reaction (25). DRH—H0 and DRD—H0 stand for the distances between appropriate atoms relative to the corresponding distances in the activated complex. For transparency, selected quantities have been rescaled by: aDE = 101, a# = 3  101. Method: DFT/ B3LYP/aug-cc-pVTZ/NAO.

5.2. Dihydrogen interaction and the stability of the biphenyl single bond rotation In the last few years, the problem of the influence of dihydrogen interaction –H  H– on the stability of the biphenyl (C12H10, Phe– Phe, depicted in Fig. 3) single bond rotation has become the subject of heated debate in a scientific literature (e.g. [44–46]). Structural and energy analyses of biphenyl rotamers testify that at the equilibrium geometry the dihedral angle between two phenyl rings is u  \C3–C1–C2–C4  44° [47] while the local energy maximum one can observe for u  0°.

Fig. 3. Equilibrium geometry of the planar biphenyl molecule with featured atomic interactions significantly contributing to the chemical bond Phe–Phe. Method: DFT/ B3LYP/6-311g**.

76

D.W. Szczepanik, J. Mrozek / Computational and Theoretical Chemistry 1026 (2013) 72–77

In order to identify atomic interactions responsible for the torsion of biphenyl molecule we have performed constrained geometry optimizations for 18 different rotamers of C12H10 with the dihedral angle u varying from 0.0° to 90.0° by 5.0° and using DFT/B3LYP/6-311g** method. Total energies, Wiberg bond orders and bond correlation coefficients (23) has been evaluated separately for each rotamer using DFT/B3LYP/aug-cc-pVTZ/NAO method. The comparison of total energies calculated for planar and equilibrium geometry of biphenyl allows one to estimate the energy stabilization of the latter against the former for about 1.8 kcal mol1 which reasonably well reproduces the experimental value of 1.5 kcal mol1 [47]. Fig. 4 depicts the dependence between u and the relative shares of Wiberg indices of particular atomic interactions (detailed in Fig. 3) in the overall bond order of Phe– Phe:

DW A—B ¼

ðuÞ W A—B ðuÞ W Phe—Phe



ðEQÞ W A—B ðEQÞ W Phe—Phe

!  100%;

ð27Þ

where A and B label atoms from two different phenyl rings. Even a cursory analysis of values DWA–B in Fig. 4 allows one to identify only one interaction that monotonically decreases as the dihedral angle u increases, and assumes negligible values for u > 41.1 (calculated equilibrium geometry), i.e. the dihydrogen –H  H– interaction. Thus, the dihydrogen bond seems to be a natural candidate for interaction determining the torsion of biphenyl molecule. However, since Wiberg bond order is a quadratic measure of electron population involved in the atom–atom (or fragment–fragment) interaction, it does not allow one to evaluate both, the (absolute) bonding/antibonding character of this interaction as well as the (relative) influence of this electron population on other chemical bonds in the molecule. In order to prove beyond reasonable doubt that dihydrogen bonds evinces destabilizing influence on the Phe–Phe single-bond rotamers from planar to the equilibrium one, we have performed the correlation analysis between selected dihydrogen bond (–H13  H14–) and the C1–C2 bond which tends to cover up to 85% of the overall Phe–Phe bond covalency. Fig. 5 presents the relative total energy (DE), Wiberg bond order (W H13 —H14 ) and the bond-correlation coefficient (#C1 —C2 jH13 —H14 ) as a function of dihedral angle u. For comparison, the Yamasaki–Goddard bond-correlation coefficients [48,49] based on the purely statistical approach have been depicted. By Yamasaki and Goddard, the coefficient of correlation between two chemical bonds in question is an appropriately normalized mean value of the covariance of bond-order operators BC1 —C2 and BH13 —H14 :

cC1 —C2 jH13 —H14 ¼ b0

D   E BC1 —C2  hBC1 —C2 i  BH13 —H14  hBH13 —H14 i ;

ð28Þ

e Fig. 4. Functional dependence of the relative bond contributions (27) featured in Fig. 3 on the dihedral angle u. Method: DFT/B3LYP/aug-cc-pVTZ/NAO.

Dihedral angle Fig. 5. Functional dependence of the relative total energy DE in (kcal mol1), the Wiberg-type bond order W H13 —H14 and bond-correlation coefficients #C1 —C2 jH13 —H14 and cC1 —C2 jH13 —H14 on the dihedral angle u. For transparency, selected quantities have been rescaled by: aDE = 2  101, aW = 102, a# = ac = 104. Method: DFT/B3LYP/augcc-pVTZ/NAO.

where the normalization constant b0 assures that 1 6 cC1 —C2 jH13 —H14 6 1. More technical details about this definition can be found in [48,49]. As follows from Fig. 5, the functional relationship between dihedral angle and both bond-correlation descriptors evidently confirms our presumption about the local destabilizing influence of dihydrogen bond in the biphenyl molecule. Within the range of 0.0° 6 u 6 41.1° both coefficients, #C1 —C2 jH13 —H14 and cC1 —C2 jH13 —H14 assume negative values while at the equilibrium geometry point they are very close to zero (approaching the limit of the established numerical accuracy). For u > 41° the dihydrogen interaction practically vanishes (DWA–B ? 0) and hence the observed positive bond-correlation is meaningless. It has to be emphasized that the essential advantage of using bond-correlation coefficient (23) rather than (28) comes from the fact that calculation of the former involves only second-order conditional probabilities within the reduced atomic resolution, while the latter is determined directly by the elements of the density matrix D. Thus, the use the definition (28) is limited only to idempotent density matrices. Furthermore, as demonstrated in [48], the sign of the Yamasaki–Goddard bond-correlation coefficient is determined by symmetrized P fourth-order terms of type A;B;C;D Di;j Dj;i0 Di0 ;j0 Dj0 ;i . Hence, in the case i;j;i0 ;j0 of more accurate wave function calculations involving large basis set the computational cost of evaluating cA–BjC–D is incomparably higher then the corresponding cost of calculation #A–BjC–D. In particular, the value of #C1 —C2 jH13 —H14 was calculated about 250 times faster than the corresponding value of cC1 —C2 jH13 —H14 (within the adopted basis set the overall number of basis functions centered on atoms C1, C2, H13 and H14 was 160). In conclusion, the results of the bond correlation analysis involving the formalism of through-space and through-bridge interactions looks quite promising. Proposed in this paper purely-probabilistic formalism defines the correlation between two chemical bonds as determined by the difference between appropriately normalized fourth-order joint probabilities corresponding to direct (through-space) and indirect (through-bridge) communication within a four-state time-homogeneous Markov chain. Alternative bond-correlation coefficient introduced earlier by Yamasaki and Goddard is based on the concept of the hierarchy of statistical covariance operators. The proposed correlation coefficient requires computing only the second-order probability terms when working within the condensed-atomic resolution and thus they are far less computationally expensive making them suitable for correlation analysis of chemical interactions between large molecular fragments. In the nearest future we are going to carry

D.W. Szczepanik, J. Mrozek / Computational and Theoretical Chemistry 1026 (2013) 72–77

out a more insightful investigation of properties of both bond-correlation measures and possibly implement our method in one of the commonly used quantum-chemistry software. Acknowledgement This work was supported by the T-Donation for Young Scientists and PhD Students, Grant No. PSP: K/DSC/001469 (from Department of Chemistry, Jagiellonian University). References [1] [2] [3] [4] [5] [6] [7]

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