Equivalence between the Classical and Quantum IR Spectral Density

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On the Equivalence between the Classical and Quantum IR Spectral Density Approaches of Weak H-Bonds in Absence of Damping Najeh Rekik, Jamal Suleiman, Paul Blaise, and Marek Janusz Wojcik J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b00269 • Publication Date (Web): 13 Feb 2018 Downloaded from http://pubs.acs.org on February 14, 2018

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The Journal of Physical Chemistry

On the Equivalence between the Classical and Quantum IR Spectral Density Approaches of Weak H-Bonds in Absence of Damping Najeh Rekik a b

a,b;

, Jamal Suleiman a , Paul Blaise c and Marek J. Wojcik

d

Physics Department, Faculty of Science, University of Ha’il, Kingdom of Saudi Arabia. Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada. (*) E-mail: [email protected]

c

Laboratoire de Mathématiques et Physique (LAMPS), 52 Av. Paul Alduy, Université de Perpignan Via Domitia (UPVD), 66860 Perpignan Cedex, France d

Laboratory of Molecular Spectroscopy, Faculty of Chemistry, Jagiellonian University, 30-387 Krakow, Gronostajowa 2, Poland

February 13, 2018

Abstract

The aim of this paper is to overhaul the quantum elucidation of the spectral density (SD) of weak H-bonds treated without taking into account any of the damping mechanisms. The reconsideration of the SD is performed within the framework the linear response theory. Working in the setting of the strong anharmonic coupling theory and the adiabatic approximation, the simpli…ed expression of the classical SD, in absence of dampings, is equated to be taken R by ICl (!) = Re 01 GCl (t)e i t dt in which the classical-like autocorrelation funco n tion (ACF), GCl (t); is given by GCl (t) = tr ( ) f (0)g f (t)gy . With this consideration, we have shown that the classical SD is equivalent to the lineshape obtained by F (!) = ICl (!), which in turn is equivalent to the quantum SD R given by IQu (!) = Re 01 GQu (t)e i t dt , where GQu (t) is the corresponding quano n R tum ACF having for expression GQu (t) = 1 tr f (0)g f (t + i ~)gy d . Thus, we 0 have shown that for weak H-bonds dealt without dampings, the SDs obtained by the quantum approaches are equivalent to the SDs geted by the classical approach in which the incepation ACF is however of quantum nature and where the lineshape is the Fourier transform of the ACF times the angular frequency. It is further shown that the classical approach dealing with the SD of weak H-bonds leads identically to the result found by Maréchal and Witkowski in their pioneering quantum treatment where they ignored the linear response theory and dampings. ACS Paragon Plus Environment 1

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Introduction Because it is crucial to all life on Earth, hydrogen bonds (H-bonds) are viewed as

the most important interactions

1 5

. They are responsible for water’s unique solvent

features 6 9 . Several chemical and biological processes are imposed by H-bonds. For example, the double helices of DNA are together hold by H-bonds and without H-bonds, DNA would have to exist at di¤erent structures

10 12

. H-bonds are also responsible for

determining the three-dimensional structure of proteins and even considered as weak interaction compare to other type of bonds, H-bonds are strong enough to keep things together 13 15 . From a spectroscopic standpoint, the presence of H-bonds introduce several e¤ects on the spectral density (SD) of the IR hydrogen stretching band

16 23

S

! H

X

stretching mode by comparison to free

. For example, one may observe a strong frequency shift

towards lower values, an extensive broadening of the bandshape 24 29 , a strong enhance of the peaks intensity 24; 26; 30 35 , dramatic changes due to the isotopic e¤ect 36; 37 , and a similarity of the spectral envelope in the gas and condensed phases are often observed 38; 39

, etc...Thus, it is necessary to erect a general theory that can capable to elucidate

all these dramatic changes in the IR spectrum. This theory may shed lights on the comprehension of the H-bond dynamics. Over the last decades, many theories have been performed and a much interest to theoretical studies of H-bonds has been elucidated 40 45

.

In addition to classical and semi-classical theories dealing with the features of the IR ! lineshapes of the S X H stretching mode 46 57 , more performing quantum theories 17; 19; 21 26

have been realized. Most of the classical and quantum theories of H-bonds

dealing with the IR SD of the high stretching frequency mode,

S

X

! H , are treated

within the linear response theory 58 60 according to which the SD is the Fourier transform of the autocorrelation function (ACF) of the dipole moment operator of the high frequency mode. Furthermore, most of the classical and quantum theories have been conducted within the framework of the strong anharmonic coupling theory 35

43; 61

. Within

the setting of this theory, the fast mode is anharmonically coupled to the slow mode, and a linear dependence of the frequency of the fast mode on the coordinate of the slow mode is generally assumed

42; 43

can be well performed

. In case we treat weak H-bond, the adiabatic approximation

62; 63

. This approximation enables us to segregate the fast mo-

tion of the high frequency mode from the slow one of the H-bond bridge. Maréchal and Witkowski (MW) 62 have introduced and employed the adiabatic approximation for bare H-bonds (monomers) for the …rst time. Their physical standpoint was to consider that the slow mode is frugally an harmonic oscillator when the fast mode is in its fundamental state and becomes abruptly a driven harmonic oscillator when the fast mode is in its …rst excited state. This consideration leads to describe the harmonic oscillator characterizing ACS Paragon Plus Environment 2

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the slow mode by e¤ective Hamiltonians. The nature of these Hamiltonians depends on !

the degree of excitation of the fast mode. Consequently, the IR SD of the S X H mode is merely viewed as a cumulating result of the excitation of the fast mode and of all permitted transitions between the energy levels of the two H-bond bridge oscillators corresponding, respectively, to the ground and …rst excited states of the fast mode. The SD is then a set of Dirac delta peaks leading to the Franck-Condon progression

62

.

We propose here to compare the spectral densities (SDs) obtained by classical and quantum theories of hydrogen bonded complexes susceptible to have in their structure weak H-bonds. We start from the model dealing with the quantum theory of weak H-bond in the absence of relaxation

64 66

which lead to the Dirac delta peaks Franck-

Condon progression. The Hamiltonian of the weak H-bond is written within the adiabatic approximation. Using some canonical transformations, we pass to non-Hermitean Hamiltonian operators that are useful for the description of the SD of weak H-bonds by a quantum model. The excitation of the fast mode allows us to describe the slow mode by a non-Hermitean Hamiltonian having the expression of a driven quantum harmonic oscillator. Based on this description, it is now allowed to obtain the time dependence of the dipole moment operator of the high frequency mode via the Heisenberg transformation involving the obtained non-Hermitean e¤ective Hamiltonians and consequently get the full ACF of the dipole moment operator. By aid of the linear response theory, we than come up with the SDs by Fourier transform of the ACF. The SDs obtained in this way, will appear to …t the classical approach having an ACF of quantum nature and where the SD is the Fourier transform of the ACF times the angular frequency. The results of this treatment therefore shed light on equivalence between the classical and the quantum approaches dealing with weak H-bonds in absence of relaxation mechanisms. This treatment provide us with a very simple, general and ‡exible theoretical tools allowing for quantitative interpretations of the SDs in terms of the di¤erent mechanisms susceptible to intervene in the area of weak H-bonds within the strong anharmonic coupling theory and the adiabatic approximation.

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Theoretical considerations. The classical approaches dealing with lineshape of weak H-bond and working within

the linear response theory assumed that the

! H::: Y

X

stretching mode of the H-

bond can be represented by stochastic process, and suppose a lost of the phase coherence due to the coupling between the two vibrational modes 46

57

S

X

! H

: The spec-

tral density (SD), ICl (!), analyzed by these classical approaches is obtained by Fourier transform of the autocorrelation function (ACF) GCl (t) of the dipole moment operator 58; 59

: ICl (!) = Re

Z

1

GCl (t)e

i t

dt

(1)

0

It is important to note that in these approaches, the ACF is written in its usual classical expression despite the fact that the calculations are of quantum nature. The ACF, GCl (t), may be written with the following equation: GCl (t) = tr

n

y

( ) f (0)g f (t)g

o

(2)

Here, f (0)g is the dipole moment operator at initial time and f (t)g the same operator at time t. ( ) is the Boltzmann density operator and

is the statistical parameter which

is related to the absolute temperature T via the Boltzmann constant kB according to: = 1=kB T

Here, the lineshape obtained by the classical approaches is proportional to the SD times the angular frequency

48 50

: F ( ) = ICl (!)

(3)

Besides, in the quantum approaches, the SD is considered to be the Fourier transform of an ACF which must be of quantum nature. IQu (!) = Re

Z

1

GQu (t)e

i t

dt

(4)

0

This ACF is given by the expression: GQu (t) =

1

tr

(

( )

Z

0

y

f (0)g f (t + i ~)g d

)

(5)

where ~ is the Planck constant. The purpose of the present paper is to show that, in abR sence of damping, the classical SD obtained by: ICl (!) = Re 01 GCl (t)e i t dt ; in which the classical-like autocorrelation function (ACF) is given by GCl (t) = tr gives rise to lineshapes that can be obtained by: F (!) = equivalent to the quantum SD given by IQu (!) = Re

R1 0

4

y

( ) f (0)g f (t)g

o

,

ICl (!), which in turn are

GQu (t)e

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n

i t

dt .

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3

Lineshapes

3.1

S

! H within the linear response theory.

X

The Quantum ACF of weak H-bonds We shall explore the lineshape of the

X

S

! H

mode of weak H-bonds within the

scope of the linear response theory. Most of the quantum approaches dealing with the ! IR spectra of the S X H mode of weak H-bonds assume that both of the fast and slow mode are taken to be harmonic

21; 22; 31

. Following MW

62

, we shall consider the

Hamiltonian of the fast mode to be harmonic like with respect to the space coordinate but with a dependence of the angular frequency of this mode on the coordinate position of the harmonic slow mode. The notations involved in the expressions of the elementary quantum harmonic oscillators of the di¤erent vibrational modes intervening in the present approach are given in Table 1. With these notations, the total Hamiltonian HT ot of the weak H-bond is considered to be the sum of HF ast and HSlow , the Hamiltonians of the fast and slow modes, respectively. Thus, HT ot may be written with the following expression 20; 39; 60

: HT ot = HF ast + HSlow

(6)

HSlow = ay a+ 12 ~!

(7)

Where

HF ast = by b+ 12

~! ay ; a

(8)

The angular frequency of the fast mode ! ay ; a is expanded up to …rst order with respect to the Bosons of the slow mode according to the strong anharmonic coupling theory. With this assumption, one has: ! ay ; a = ! +

ay + a !

(9)

Now, let us consider the eigenvalue equations of the Hamiltonians of the fast and slow modes. They are respectively: k+

1 2

jfkgi

(10)

HSlow j(m)i = ~! m +

1 2

j(m)i

(11)

HF ast jfkgi = ~!

We shall now, consider the ACF of the dipole moment operator given by Eq.(2). The dipole moment operator of the fast mode (0) involved in Eq.(2) is given by: (0) =

01

jf0gi hf1gj

(12)

in which appear the ground state jf0gi and the …rst excited state jf1gi of the fast mode. Moreover, at time t the dipole moment operator is given by the Heisenberg transformation

19; 21

: (t) =

01

exp

it ~ HT ot

jf0gi hf1gj exp


it ~ HT ot

(13)


Besides, the Boltzmann operator

B

B

Page 6 of 19

; appearing in Eq. (2) and Eq. (5), is given by:

( )_

1 e ZT ot

HT ot

(14)

where Z is the partition function. According to Eq. (5), the ACF may be written in its …nal form with the following expression: G(t) =

3.2

2 10

ZT ot

2

tr 4

HT ot

e

R

0

jf0gi hf1gj 1f1g exp i ~

hf0gj exp

i ~

(t + i ~) HT ot

(t + i ~) HT ot

d

jf1gi

3 5

(15)

The ACF within the adiabatic approximation

Now, let us focus on the ACF within the adiabatic approximation. We recall that within the strong anharmonic theory rationalized by MW 62 , the Hamiltonian of the low and high frequency modes is given by: HT ot =

p2 1 P2 + + m! 2 (Q) [q 2M 2m 2

2

qe (Q)] + 21 M ! 2 Q2

(16)

Here, the angular frequency of the fast mode !(Q) and the equilibrium position qe (Q) are assumed to be a function of the slow mode coordinate. The modulation of the angular frequency of the fast mode !(Q) by the hydrogen bond bridge Q, up to …rst order is given by: !(Q) = ! + b Q

(17)

Besides, we shall neglect the dependence of the equilibrium position on the hydrogen bond bridge Q and restricting the calculations to …rst order of !(Q). Indeed, MW 62 have shown that the consideration of the only parameter (b) of the Eq. (17) and the neglect of qe (Q) allow to reproduce main characteristics of the envelope of spectra. Thus, owing to the above equations and considerations, the full Hamiltonian HT ot of weak hydrogen systems can be written by the following form: HT ot

p2 1 + m! 2m 2

=

2 2

q

+

P2 1 + M ! 2 Q2 2M 2

(18)

1 +bm! q 2 Q+ mb2 q 2 Q2 2

Let us now perform the adiabatic approximation

62; 63

, which allow to separate the

motions of the high frequency mode from the slow H-bond bridge. Within this approxifkg mation, the Hamiltonian of the bridge may be described by e¤ective Hamiltonians H . These e¤ective Hamiltonians are given by fkg

H

=

67 69

:

P2 1 + M ! 2 Q2 +kbQ + k~! 2M 2

(19)

Recall that the nature of the e¤ective Hamiltonians changes with the excitation degree (k) of the high frequency stretching mode. Here, the fundamental state and the …rst excited state of the high frequency stretching mode are corresponding to (k = 0) and ACS Paragon Plus Environment 6

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(k = 1), respectively. Furthermore, instead of the strong anharmonic coupling parameter (b) of the Eq. (17), we shall use the dimensionless one given by: b = !

r

~ 2M !

(20)

It is important to note that the last parameter ( ) is the same parameter that appears in Eq. (9). By using the Boson representation and after introducing some canonical transformations, we can easily show that the e¤ective Hamiltonians characterizing the low frequency stretching mode within the quantum representation labelled II given by the following equation: fkg

H

II

= ay a+ 21

ay + a

~! + k

2

67 69

, are

~! + ~!

(21)

Here, it is important to distinguish that the e¤ective ground state Hamiltonian (fkg = 0) is that of a simple quantum harmonic oscillator whereas the e¤ective …rst excited Hamiltonian (fkg = 1) is that of a driven quantum harmonic oscillator. Within the quantum representation II

67 69

G(t) =

, it is now possible to rewrite the ACF (Eq.(5)) according to:

2 10 f0g

ZII

2

f0g

HII

e

tr 4

o 3 n f1g i (t + i ~) H jf1gi jf0gi hf1gj exp II ~ 0 5 n o f0g i hf0gj exp d (t + i ~) H II ~

R

(22)

Thus, it may be suitable to use for the ACF of the dipole moment operator (Eq.(22)) a more simpli…ed expression. For this purpose, we shall use the quantum representation labelled fIIIg 67 69 , in which the e¤ective Hamiltonians characterizing the H-bond bridge become diagonal. This consideration may be implemented by aid of the following canonical transformation: fkg

H

where

III =

n fkg o fkg n fkg o+ A (k ) HII A (k )

n fkg o A (k ) = exp k

ay

After some rearrangement, the expression of H fkg

H

ay a+ 12 ~!

III =

a fkg

III

k (k + 1)

jfkgi hfkgj

(23)

(24)

(Eq. (21) and Eq. (23)) becomes: 2

~! + k~!

(25)

Owing to the above equations, the basic starting ACF takes, in view of Eq. (5), the following form: G(t) =

2 10 f0g

ZIII

2

tr 4

H

e

3 f1g jf0gi hf1gj exp ~i (t + i ~) HIII jf1gi 5 n o f0g i (t + i ~) H hf0gj exp d III ~

f0g III

R

0

(26)

by inserting the identity operator (I):

o n fkg o n fkg o+ n Ifkg = A (k ) A (k )

(27)

the ACF (Eq.(26)) may be written by the following expression: G(t) =

2 10 f0g

ZIII

2

tr 4

e

H

f0g III

R

0

3 o n f1g jf0gi hf1gj If1g exp ~i (t + i ~) HIII If1g jf1gi 5 o n f0g i If0g d hf0gj exp ~ (t + i ~) HIII


(28)


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To simplify the last expression of G(t), we …rst perform the action of the operators involved in the canonical transformation. Then, after e¤ecting a circular permutation of the operators involved in the trace and implementing the properties of the e¤ective operators, we get: G(t)=

2 10 f0g

ZIII

2

tr 4

exp

n o 3 f1g f1g jf0gi hf1gj A ( ) exp ~i (t + i ~) HIII 5 o n f0g f1g i (t + i ~) H d A ( ) + jf1gi hf0gj exp III ~

n

f0g

HIII

oR

0

(29)

On the other hand, one may introduce in the ACF (29) in the quantum representation {III } G(t)=

2 10

Z

2

tr 4

e

(ay a+ 12 )~! P R Af1g ( ) 0 n f1g A ( ) exp

+

j(n)i h(n)j exp

i ~

n

i ~

f1g

(t + i ~) HIII

o f1g (t + i ~) HIII d

o 3 5

(30)

Here, Z is the partition function of the slow mode harmonic oscillator. It is given by: h Z = tr e

i

(ay a+ 12 )~!

Performing the trace, the ACF takes the form: G(t)

=

2 10

XX

Z exp

i2

1 e (m+ 2 )

2

~!

2

jAmn ( )j

! (t + i ) exp

2

i2

(31)

Z

exp [i (n

m) ! (t + i )]

(32)

0

! (t + i ) d

where Amn ( ) are the matrix elements of the translation operator involved in Eq.(29): f1g

jAmn ( )j = h(m)j A

( ) j(n)i

(33)

These matrix elements are the Franck-Condon factors appearing in the pioneering work by MW 3.3

62

.

Integration over By integration over

G(t)

=

2 10

Z

XX

[(m

in the ACF one obtains: 2

jAmn ( )j exp fi (n

1 n) + 2 2 ] ~!

~!

m) !tg exp fi! tg exp

exp

2

2

n ~!

i2

~!

2

!t

exp

(34) m+

1 2

~!

Now, we may observe that, even at room temperature one has: ~! >> kB T

As a consequence, since ! >> ! , we have for all n: exp

2

2

n ~!

w0

~!

Thus, the above ACF may be approached very satisfactorily by: G(t)

=

2 10

Z

XX

1 e (m+ 2 )

exp fi! tg exp

i2

~!

2

2

jAmn ( )j exp fi (n

!t

[(n

m)


m) !tg

1 2 2 ] ~! + ~!

(35)

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3.4

Fourier transform of the integrated ACF Now, performing the Fourier transform on the ACF given by Eq.(35), we get: F( )

XX

2 10

=

Z

1 e (m+ 2 )

(n

m)

~!

2

2

1 2 2 ] ~! + ~!

2

jAmn ( )j

[(n

m)

(36)

!+!

Using the properties of the so-called Dirac delta ( ) function argument, the last equation giving rise to the Fourier transform, F ( ), becomes: F( )

2 10

=

Z

XX [(n

(n

m)

2

2

~! + ~!

1 2 2 ] ~! + ~!

m)

1 e (m+ 2 )

(n

m)

2

~!

2

jAmn ( )j

2

!

(37)

!

Thus, after simpli…cations, to: F( ) =

2 10

Z

XX

1 e (m+ 2 )

~!

2

jAmn ( )j

(n

m)

2

2

!

!

(38)

In this equation, one may recognize the Franck-Condon progression appearing in the model of MW

62

dealing with weak H-bonds. Thus we may conclude that the quantum

and the classical spectral densities approaches of weak hydrogen bonded complexes in absence of relaxation mechanisms are equivalent since we do have: Re

Z

0

1

tr

n

y

( ) f (0)g f (t)g =

2 10

Z

XX

e

o

e

i t

dt

=

1

Re

"Z

1

tr

0

(m+ 21 )~! jA ( mn

2

)j

( Z

y

0

h

!

(m

f (0)g f (t + i ~)g d n) !

2

2

!

i

)

e

i t

dt

(39)

In Figure 1, we report a simple ‡owchart that can facilitate understanding and coding the approaches involved in this paper. Furthermore, it is important to note that presented approaches can be used in quantifying the role of dynamics in enzyme catalysis. Recent pioneering works 70; 71 have shown that the proton transfer reactions in solution and enzymes obey the transition state theory. Within the framework of this theory many experimental observations for a certain number of enzymes are reproduced 70 . Finally, it must be underlined that the growing interest in H-bonds, especially in biological systems should be usefully accompanied by the development of a theoretical tool in the …eld of vibrational spectroscopy. The approaches presented herein may contribute towards understanding vibrational nature of olfaction mechanism

72; 73

and receptor activation

via studying the quantum nature of drug-receptor interactions since the deuteration of H-bonds changes the binding a¢ nities for histamine receptor ligands 74 .


#


4

Conclusion In this paper, we have overhauled the quantum elucidation of the spectral density

of weak H-bonds without taking into account any of the relaxation mechanisms. This quantum elucidation is performed within the framework of the linear response theory. Working within the scope of the strong anharmonic coupling theory, and in the setting of the adiabatic approximation, we have shown that the spectral density developed in the previous quantum approaches dealing with weak H-bonds is equivalent to the more general classical one in which the inception ACF is of quantum nature and where the bandshape is given by the Fourier transform of the ACF times the angular frequency. More precisely, we have shown that the classical approach performed to weak H-bonds is then simple one and leads to the quantum result of Maréchal and Witkowski who ignored in their treatment the linear response theory. Furthermore, it must be underlined to note that there are several approaches that can be used to compute the IR spectral densities, such as molecular dynamics simulation on the Car-Parrinello level, ab-initio methods, and hybrid QM/MM. These molecular dynamics simulations are, however, computationally expensive, in contrast to the approaches presented in this paper.

Acknowledgment: Dr. Najeh Rekik thanks the Deanship of Scienti…c Research, University of Ha’il, Kingdom of Saudi Arabia, for the …nancial support under grant number: 0160747. Prof. Wojcik thanks the National Science Center, Poland for the …nancial support under grant number: 2016/21/B/ST4/02102 Previous address: Dr. Najeh Rekik is on leave from Laboratoire de Physique Quantique, Faculté des Sciences de Monastir, Route de Kairouan, 5000 Monastir, Tunisia.


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Table 1: De…nition of some notations associated with the elementary quantum harmonic oscillators. Fast mode

Parameters

! H

X

Slow mode

Y

X

Angular frequency

!

!

Reduced mass

m

M

Coordinate position

q

Q

p

P

Conjugate momentum

y

b ;b

ay ; a

by ; b = 1

ay ; a = 1

Creation and annihilation operators Commutation rule Dimensionless position coordinate

q=

Conjugate moment

p=

Basis notation

! Y

H

p1 2 i p 2

by + b

Q=

by

P =

jfkgi

b

p1 2 i p 2

ay + a ay

a

j(m)i

Figures caption: Figure 1: A simple ‡owchart explaining the equivalence between the classical and quantum IR spectral density approaches of weak H-bonds in absence of damping.

Table caption: Table 1: De…nition of some notations associated with the elementary quantum harmonic oscillators.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Weak hydrogen bonds


Absence of relaxation mechanisms

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Linear response theory

ACS Paragon Plus Environment Figure 1: A simple flowchart explaining the equivalence between the classical and quantum IR spectral density approaches of weak H-bonds in absence of damping.

Page 19 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


Weak hydrogen bonds

Absence of relaxation mechanisms

Adiabatic approximation

Quantum approaches

Linear response theory

Equivalence between IR spectral densities

TOC Graphic ACS Paragon Plus Environment

Quantum Classical approaches approaches

Equivalence between the Classical and Quantum IR Spectral Density

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