Research Article Using the Intramolecular Contribution to ...downloads.hindawi.com › archive › 2015 › 701386.pdf · Symmetry Breakdown in Molecular Crystals ... the dynamic

Research ArticleUsing the Intramolecular Contribution tothe Second Moment of NMR Line Shape to Detect SiteSymmetry Breakdown in Molecular Crystals

Ferid Bashirov1 and Nail Gaisin2

1Department of General Physics Institute of Physics Kazan Federal University Kremlevskaya Street 18Kazan 420111 Tatarstan Republic Russia2Department of Physics Institute of Petroleum Chemistry and Nanotechnology Kazan Technological UniversityKarl Marx Street 66 Kazan 420015 Tatarstan Republic Russia

Correspondence should be addressed to Ferid Bashirov fbashirmailru

Received 15 September 2015 Revised 15 November 2015 Accepted 29 November 2015

Academic Editor Guang Zhu

Copyright copy 2015 F Bashirov and N Gaisin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new approach to simulating the intramolecular contribution to the anisotropic secondmoment of NMR spectral lines broadenedby magnetic dipole-dipole interaction of nuclei is suggested The extended angular jump model is used by approximating the localhindered molecular motion (HMM) The theoretical result allow describing the site symmetry distortion by new experimentalparameters 119902

120572 the dynamic weights of irreducible representations of the HMM crystallographic point symmetry group The

application of the theory to describing the intraionic second moment of the proton NMR spectral line in monocrystallineammonium chloride proves the tetragonal distorted tetrahedral site symmetry of ammonium ions

1 Introduction

The second moment of the NMR spectral line broadeneddue to the magnetic dipole-dipole interaction has beenyielded by Van Vleck as one of the main probes of hinderedmolecular motion (HMM) in crystalline substances [1] Asthe magnitude of the second moment depends rigorouslyon geometrical and physical peculiarities of nuclear motionmeasuring the second moment allows one to test variousmodels of HMM taking place in condensed matter

At present three principal model diversities of localHMM exist the rotational diffusion model (RDM) the fixedangular jumpmodel (FAJM) and the extended angular jumpmodel (EAJM) [2 3] Using RDMshown to be very successfulby describing HMM happened in liquids [4ndash7] FAJM workssufficiently well in isotropic media of powders [8ndash10] Atlast the EAJM is appropriated for exploring the HMM ofsymmetrical molecules in any condensed molecular mediasingle crystals powders and liquids It was fruitfully appliedto discuss the broadened lines of incoherent scattering of

neutrons dielectric and infrared absorption Raman scat-tering of light and the rates of nuclear magnetic relaxationin mono- and polycrystalline molecular media [2 3] Thispaper is devoted to expanding EAJM approach to simulatethe second moment of NMR-absorption line broadeneddue to the magnetic dipole-dipole interaction in condensedmolecular media It is dedicated also to showing that theintramolecular part of the second moment is sensitive tosymmetry breakdown effect in the HMM As an applicationwe revise the experimental data of Bersohn and Gutowskyconcerning the 2nd moment of the proton spectral linein the single crystal of ammonium chloride [11] A newinterpretation of the data follows the tetragonal distortedsite symmetry of NH

4

+ ions in NH4Cl that firstly has been

determined by the proton relaxation experiments [12]

2 Theory

According to Van Vleck the contribution of identical res-onant nuclei to the second moment of NMR spectral line

Hindawi Publishing CorporationInternational Journal of SpectroscopyVolume 2015 Article ID 701386 6 pageshttpdxdoiorg1011552015701386

2 International Journal of Spectroscopy

broadened owing to their magnetic dipole-dipole interactionwith resonant andwith nonresonant nuclei in a crystal latticecan be calculated by using a well-known expression [1]

119872(119894)

2= sum

119895

119872(119895119894)

2= sum

119895

3

41205742

119895ℏ2

119868119895(119868119895+ 1)

(3cos2120579119895119894

minus 1)2

1199036

119895119894

(1)

where the index 119894 labels a target resonant nucleus of a chosenmolecule and the index j other nuclei of the substance 120574

119895and

119868119895are respectively the gyromagnetic ratio and the quantum

number of the nuclei ℏ is Planckrsquos constant divided by 2120587and 120579

119895119894is the polar angle formed by the internuclear vector

r119895119894with the induction vector of the stationary magnetic field

B0determined in the laboratory reference frame (LRF)For convenience we shall replace (3cos2120579

119895119894minus 1)2

by (161205875)|119884(2)

0(120579119895119894)|2

= (161205875)|119884(2)

0(120593119895119894 120579119895119894)|2 where

119884(2)

0(120579119895119894) = 119884

(2)

0(120593119895119894 120579119895119894) is the zeroth element of normalized

spherical harmonic of the second order the zerothcomponent of the unit spherical tensor of the 2nd rankwhich depends on the polar angle 120579

119895119894 but it has no

dependence of the azimuth angle 120593119895119894 Moreover we shall

rewrite (1) in terms of spherical tensor components as

119872(119895119894)

2= sum

119895

1198722119895119894

= sum

119895

12120587

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

10038161003816100381610038161003816119884(2)

0(120593119895119894 120579119895119894)10038161003816100381610038161003816

2

(2)

By the way the angles 120579119895119894and120593119895119894relate LRFwhereas the inter-

nuclear vectors are fixed in the crystallographic referenceframe (CRF) Taking into account the transformation rule ofspherical tensors by a rotation to the three-dimensional Eulerangles (120601 120599 120585) from CRF to LRF given by the expression

119884(2)

0(120593 120579) =

2

sum

119898=minus2

119863(2)

0119898(120601 120599 120585) 119884

(2)

119898(1205931015840

1205791015840

) (3)

where 119863(2)

0119898(120601 120599 120585) is an element of Wigner matrix we can

express 119872(119895119894)

2in the form

119872(119895119894)

2= sum

119895

12120587

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

2

sum

119898=minus2

119863(2)

0119898(120601 120599 120585) 119884

(2)

119898(120593119895119894 120579119895119894)

1003816100381610038161003816100381610038161003816100381610038161003816

2

(4)

Replacing 119863(2)

0119898(120601 120599 120585) = radic(41205875)119884

(2)

119898(120599 120601) allows us to

reduce (4) to

119872(119895119894)

2= sum

119895

48120587

251205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

2

sum

119898=minus2

119884(2)

119898(120601 120599) 119884

(2)

119898(120593119895119894 120579119895119894)

1003816100381610038161003816100381610038161003816100381610038161003816

2

(5)

where (120601 120599) and (120593119895119894 120579119895119894) are respectively the spherical

angles of the vector B0and the internuclear vector r

119895119894

determined with respect to CRFAt sufficiently low temperatures a regime of rigid lattice

is valid in molecular crystals At high temperatures (T gt

40K) the direction of some internuclear vectors changesaccidentally due to classical rotational displacement Conse-quently angles (120593

119895119894 120579119895119894) become random functions of time

Such motion changes a spectral distribution law of NMRsignals Therefore the simulation of a second moment formolecular substances has to be performed by accounting forintramolecular nuclei as well as intermolecular ones

119872(119895119894)

2= 119872(intra119895119894)2

+ 119872(inter119895119894)2

(6)

The term 119872(inter119895119894)2

denoted by the label ldquo(inter119895119894)rdquo is theintermolecular part of 119872

(119895119894)

2 Its calculation is out of the

interest of this paperThe other term of the sum (6) 119872

(intra119895119894)2

labeled byldquointra119895119894rdquo is the part of 119872

(119895119894)

2delivered by nuclei inner with

respect to the given molecule It is convenient to present119872(intra119895119894)2

in the form [5 13 14]

119872(intra119895119894)2

=

119873minus1

sum

119895=1

3

41205742

119895ℏ2

119868119895(119868119895+ 1)int

120575]

minus120575]119869(0)

119895119894(120596) 119889] (7)

where 119873 is the number of identical resonant nuclear spins inthe considered molecule and

119869(0)

119895119894(120596) = int

+infin

minusinfin

119870(0)

119895119894(119905) exp (119894120596119905) 119889119905 (8)

is the nonnormalized spectral density function of the auto-correlation function

119870(0)

119895119894(119905) = ⟨119865

(0)

119895119894(119905) 119865(0)lowast

119895119894(119905 + 120591)⟩ (9)

In its turn the function 119865(0)

119895119894(119905) is the coordinate part of spin-

spin interaction Hamiltonian of nuclei 119894 and 119895 given by

119865(0)

119895119894(119905) = 119903

minus3

119895119894[3cos2120579

119895119894(119905) minus 1]

or 119865(0)

119895119894(119905) = (

16120587

5)

12

119903minus3

119895119894119884(2)

0[120579119895119894

(119905)]

(10)

The integration is prescribed within the limits of the doubleline width from minus120575] up to +120575] in (7) By the way 120575] =1T2 where T

2is the spin-spin relaxation time 120596 = 2120587] =

1205741198941198610is the resonance angular frequency of spins 119894 120574

119894is their

gyromagnetic ratio and 1198610is the module of induction vector

of the static magnetic fieldThe outcome of calculus by using (7)ndash(10) depends on

the physical model of molecular motion According to thegoal of the present study we shall apply EAJM approach[2 3] as the model of HMM within the framework of whichan analytical expression of the normalized spectral density

International Journal of Spectroscopy 3

function (SDF) 119869(2)

0(120596) of zero component of the 2nd rank

tensor 119884(2)

0[120601(119905) 120599(119905)] can be written for a single crystal by

119869(2)

0(120596) = 119869

(2)

0(119902120572 120601 120599 120596)

=5

2120587sum

120572

2

sum

119897=0

119902120572120591120572

1 + (120596120591120572)21198861205721198970

(120601) cos2119897120599

(11)

In agreement with the EAJM approach polar angle 120579119895119894

of the internuclear vector r119895119894does not present explicitly in

(11) Instead there are two spherical angles 120601 and 120599 which fixthe orientation of main axis of the crystallographic referenceframe in the laboratory one Index 120572 labels irreducible rep-resentations (IR) Γ

120572of the molecule motion point symmetry

group G and 119897 labels an intermediate summationTaking into consideration (11) nonnormalized SDF takes

the form

119869(119895119894)

0(120596) = 119869

(119895119894)

0(119902120572 120601 120599 120596) =

16120587

5119903minus6

119895119894119869(2)

0(119902120572 120601 120599 120596)

= 8119903minus6

119895119894sum

120572

2

sum

119897=0

119902120572120591120572

1 + (120596120591120572)21198861205721198970

(120601) cos2119897120599

(12)

The rated expressions of the factors 1198861205721198970

(120601) are tabulatedexplicitly as a function of azimuth angle 120601 for all crystal-lographic point symmetry groups of pure rotation in [23] The quantities 119902

120572are the adjustable parameters of the

theory called the dynamic weights of IR Γ120572They characterize

the symmetry distortion and have to be experimentallydetermined Parameter 120591

120572is a correlation time adapted to IR

Γ120572that relates the expression

120591120572

= (1 minus 120594minus1

120572119864sum

119894

119901119894120594120572119894)

minus1

120591 (13)

where 120594120572119894

and 120594120572119864

are the characters of 119894th and identicalclasses respectively 119901

119894are probabilities of fundamental acts

of the motion appropriate to 119894th class of the group G and 120591

is a mean time between two consequent steps of motion Thetime 120591 is ordered to Arrhenius low

120591 = 1205910exp(

119864119886

119877119879) (14)

where 119864119886is the height of activation energy barrier averaged

on the HMMmotion symmetry group and 1205910is a mean time

between two sequential attempts to overlap the barrierBy substituting (12) in (7) we shall obtain

119872(intra119895119894)2

=

119873minus1

sum

119895=1

61205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot int

120575]

minus120575]sum

120572

2

sum

119897=0

119902120572120591120572

1 + (2120587]120591120572)21198861205721198970

(120601) cos2119897120599 119889]

(15)

Performing just prescribed integration we shall finally getthe second moment expression specified by intramolecular

action of spins 119895 to spin 119894 for a given single crystal in ananalytical form

119872(intra119895119894)2

=

119873minus1

sum

119895=1

6

1205871205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599arctg (2120587 sdot 120575] sdot 120591120572)

(16)

For a powder averaging (16) over the angles 120601 (0 le 120601 le 2120587)and 120599 (0 le 120599 le 120587) reduces it to

119872(intra119895119894pow)2

= sum

119895

6

51205871205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)sum

120572

119902120572arctg (2120587 sdot 120575] sdot 120591

120572)

(17)

In a regime of the fast thermal molecular motion theinequality 2120587 sdot 120575] sdot 120591

120572≪ 1 is valid that follows the

equality arctg(2120587 sdot 120575] sdot 120591120572) = 0 In this case the dipole-

dipole contribution to the second moment disappears thatis 119872(intra119895119894)2

= 0 and so-called phenomenon of ldquospectral linenarrowingrdquo or ldquobandwidth narrowingrdquo is observed

It should be noted that in case the molecular motion isfrozen the alternative inequality 2120587 sdot 120575] sdot 120591

120572≫ 1 followed

by arctg(2120587 sdot 120575] sdot 120591120572) = 1205872 is valid Furthermore taking

into consideration that the dynamic weights of irreduciblerepresentations are normalized to unity namely sum

120572119902120572

= 1(17) reduces down to a permanent value

119872(intra119895119894pow)2

= sum

119895

3

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1) (18)

presented earlier by Abragam and Losche [15 16] Formonocrystalline samples (16) reduces to an expressionmain-taining the angular dependence in the second moment

119872(intra119895119894)2

= sum

119895

31205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599

(19)

It is noted that in the case of not resonant nuclear spinscontribute to the 2ndmoment (19) has to be corrected by thefactor 49 [17] Labelling not resonant nuclear spins by theindex 119895

1015840 this contribution will be expressed in the regime ofslow molecular motion as

119872(intra1198951015840119894)2

= sum

1198951015840

4

31205742

1198951015840ℏ2

119903minus6

11989510158401198941198681198951015840 (1198681198951015840 + 1)

sdot sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599

(20)

3 Application to MonocrystallineAmmonium Chloride

Ammonium chloride NH4Cl being one of the most studied

substances is frequently used as a touchstone of the validity


60

50

40

300

12 2

46

3 0120599 (rad) 120601 (rad)

M(in

tra)

2G

s2

Figure 1 The angular dependence of intraionic contribution to theproton NMR spectral line second moment in ammonium chloridesingle crystal drawn according to (26) for a slow motion regime ofNH4

+ cation (the numerical parameters used in (26) are 1199021= 0254

1199022= 0757 0 le 120601 le 2120587 and 0 le 120599 le 120587)

of various theories on the structure and physical propertiesof crystals It is an ionic crystal at which ammonium ionsexhibit random reorientation and consequently they have noorientation ordering Unit cell of NH

4Cl is a body-centered

cube of type bsbl At the center of a cubic cell a tetrahedronof ammonium cation is placed and its corners are occupiedby anions of chlorine The lattice parameter is 119903Cl-Cl = 119903N-N= (3844 plusmn 0024) sdot 10

minus10m four nearest protons are placedin the vicinity of a nitrogen nucleus at the distance 119903N-H =(1038 plusmn 004) sdot 10

minus10m and the neighboring protons arespaced from each other to 119903H-H = (1695plusmn004) sdot 10

minus10m [18]Below 2429 K the crystal is in its ordered phase The

random local motion of ammonium ions does not changethe ordered structure of the crystal It means that thereorientation symmetry group of any ammonium ion vectoris the point symmetry group of tetrahedron T

There are two equilibrium dispositions of NH4

+ tetrahe-dronswith equal probability in the disordered phase observedat temperatures 119879 ge 2429 K Hence any physical quantityobeys the point symmetry group of the octahedron O in thedisordered phase of NH

4Cl

It was found that both the shape of spectral line [11]and the relaxation times [12] are anisotropic and showtemperature dependence in the single crystal of NH

4Cl To

discuss the continuous wave data the quantum mechanicaltechniquewas applied [11]The relaxation datawere discussedin the framework of EAJM approach taken as the model ofclassical HMM of NH

4

+ cation [12]To examine the site symmetry of the ion NH

4

+ in theregime of its slow motion by using the second moment ofthe proton NMR spectral line in NH

4Cl we have to deal

with two contributions 119872(intraHH)2

and 119872(intraNH)2

The firstcontribution is induced by interaction of an arbitrary chosenproton with other three protons of a chosen cation (19) andthe second contributionmdashwith inner nitrogen nucleus (20)

By using 119868119895

= 119868H = 12 as the value of proton spin(119873 minus 1) = 3 as the number of adjacent protons 120574

119895= 120574H and

60

50

40

30

B 0[100]

B 0[111]

B 0[110]

3210120599 (rad)

M(in

tra)

2G

s2

Figure 2 Angular dependence of the intraionic contribution to theproton NMR spectral line second moment in ammonium chloridesingle crystal drawn according to (26) for a slow motion regime ofNH4

+ cation 120601 = 1205874 and minus1205878 le 120599 le 91205878The experimental values119872(intraexp)2

[1 0 0] = 299Gs2 (120601 = 1205874 120599 = 0) and119872(intraexp)2

[1 1 0] =52Gs2 (120601=1205874120599=1205872) [11] used by calculating the dynamicweights1199021= 0254 and 119902

2= 0757 are fitted by open circles in the graph

119903119895119894

= 119903HH we shall get from (19) the rated expression of119872(intraHH)2

as

119872(intraHH)2

=27

41205742

Hℏ2

119903minus6

(HH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599)

(21)

Substituting 1198681198951015840 = 1 120574

1198951015840 = 120574N and 119903

1198951015840119894= 119903NH in (20) follows

the rated expression of 119872(intraNH)2

119872(intraNH)2

=8

31205742

Nℏ2

119903minus6

(NH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599) (22)

Using the table data 120574H = 26753 sminus1Gsminus1 120574N =19333 sminus1Gsminus1 and ℏ = 10544 sdot 10

minus27 ergsdots [16] experimentalvalues 119903NH = 1038 sdot 10

minus8 cm and 119903HH = 1695 sdot 10minus8 cm [18]

and the theoretical expressions of the factors 1198861205721198970

(120601) 119886100

(120601)

= (18)(1 + 3cos22120601) 119886110

(120601) = minus(34)(1 + cos22120601) 119886120

(120601)

= (38)(3 + cos22120601) 119886200

(120601) = (14)(1 minus cos22120601) 119886210

(120601) =(12)(1 + cos22120601) and 119886

200(120601) = minus(14)(3 + cos22120601) [2 3] in

(21) and (22) allows us to obtain the rated explicit expressionsof 119872(intraHH)2


as

119872(intraHH)2

(1199021 1199022 120599 120601) = 283107 sdot Ω (119902

1 1199022 120599 120601) (23)

119872(intraNH)2

(1199021 1199022 120599 120601) = 11074 sdot Ω (119902

1 1199022 120599 120601) (24)

where Ω(1199021 1199022 120599 120601) the function of the dynamic weights 119902

1

and 1199022and the angles 120601 and 120599 is equal to

Ω(1199021 1199022 120599 120601) = 119902

1[(1 + 3cos22120601)

minus 6 (1 + cos22120601) cos2120599 + 3 (3 + cos22120601) cos4120599]


50

40

300 1 2 2

46

3 0120599 (rad)

120601 (rad)0 1 2 2

46

3 0120599 (rad)

120601 (rad)

20

15

M(in

traH

H)

2G

s2

M(in

traN

H)

2 G

s2

Figure 3 Surface-plot graphs of 119872(intraHH)2


the intraionic contributions to the proton NMR spectral line second moment inammonium chloride single crystal for a slowmotion regime of NH

4

+ cation drawn according to expressions (23) and (24) and 1199021= 0254 and

1199022= 0757

+ 21199022[(1 minus cos22120601) + 2 (1 + cos22120601) cos2120599

minus (3 + cos22120601) cos4120599]

(25)

By summing (23) and (24) we get the expression of fullintraionic part119872(intra)

2 which preserves the same dependence

Ω(1199021 1199022 120599 120601) of the variables 119902

1 1199022 120601 and 120599

119872(intra)2

(1199021 1199022 120599 120601) = 294181 sdot Ω (119902

1 1199022 120599 120601) (26)

The theoretical dependence of 119872(intra)2

the intraioniccontribution to the second moment of proton NMR spectralline in monocrystalline ammonium chloride prescribed by(26) allows us to calculate the numbering values of dynamicweights 119902

1and 1199022by using experimental values of 119872(intra)

2

Themean value of the secondmoment at low temperature(minus195∘C) that is in the ordered phase has been determinedfor direction of the induction vector of the external staticmagnetic field chosen along the fourfold symmetry axis inthe unit cell of NH

4Cl (B

0 [1 0 0]) equal to 365 Gs2

and along the twofold symmetry axis (B0

[1 1 0])mdash546Gs2 [11] The respective high temperature values wereequal to 66Gs2 and 26Gs2 Because of the fact that theintraionic part of the second moment decreases down tozero at high temperatures we suggest that the respectivevalues of the second moment consist of the interionic partsonly Taking into account just presented point of view aboutthe experimental second moments we can assume that thedifference between low and high temperature values of thetotal secondmoment can be considered as the pure intraionicparts for low temperature second moments Performing theappropriate calculations we get the experimental values ofthe intraionic second moments 119872

(intraexp)2

equal to 299Gs2

and 52Gs2 for directions of magnetic field induction B0

[1 0 0] and B0 [1 1 0] respectively

By substituting in (26) firstly 119872(intraexp)2

= 299Gs2 120601

= 1205874 and 120599 = 0 and then 119872(intraexp)2

= 52Gs2 120601 = 1205874and 120599 = 1205872 we get the experimental values of dynamic

weights 1199021and 1199022 1199021= 0254 and 119902

2= 0757 These values of

dynamic weights of two-dimensional and three-dimensionalirreducible representations for tetrahedron point symmetrygroup of ammonium ion motion in the ordered phase ofNH4Cl are in good agreement with our data 119902

1= 025 and

1199022= 073 derived from proton NMR relaxation experiments

earlier [12] It results in the tetragonal distorted tetrahedralsite symmetry of ammonium ions in the ordered phase ofNH4Cl It is noted that no anisotropy is predicted neither

for the intraionic second moment no for the spin-latticerelaxation rates in the ideal cubic symmetry position In thiscase 119902

1and 1199022take respective values equal to 04 and 06

A surface-plot graph of the intraionic contribution119872(intra)2

to the proton NMR second moment in ammoniumchloride single crystal for a slow motion regime of NH

4

+

cation is shown as a function of spherical angles 120601 and 120599

of the crystal orientation in the static magnetic field for theexperimental values of dynamic weights 119902

1= 0254 and 119902

2=

0757 in Figure 1 In Figure 2 the graph of 119872(intra)2

is shownfor 120601 = 1205874 as a function of the polar angle 120599 which forms theinduction vector B

0of the static magnetic field with respect

to the direction [1 0 0] of the unit cell main axis Here onecan see the values of119872(intra)

2for threemain orientations of the

external magnetic field in the unit cell of the NH4Cl crystal

B0 [1 0 0] B

0 [1 1 1] and B

0 [1 1 0]

Estimation of comparative contributions of the hydrogenspins versus nitrogen one to the intraionic 2nd momentis represented as interesting The graphs of those drawnaccording to formulas (23) and (24) and accounting for 119902

1

= 0254 and 1199022= 0757 are shown in Figure 3 It is noted

that they are spacelike in relation 119872(intraHH)2

119872(intraNH)2

=28310711074 = 25565

4 Conclusion

The presented approach of simulating the intramolecularcontribution to the second moment of NMR spectral lineallows one to perform an accurate analysis of dynamicsand geometry of internal motion It requires using theextended angular jump model by approximating the


hindered molecular motion knowledge of structure datafor the studied crystal and point symmetry property ofthe molecule motion as well as that of its surroundingsinclusively Alternatively by examining the anisotropicbehavior of the second moment this theory allows one toobtain the precise structure knowledge

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J H VanVleck ldquoThe dipolar broadening ofmagnetic resonancelines in crystalsrdquo Physical Review vol 74 no 9 pp 1168ndash11831948

[2] F Bashirov and N Gaisin ldquoThe theory of hindered molecularmotion and its application to spectroscopic studiesrdquo Crystallog-raphy Reviews vol 16 no 1 pp 3ndash87 2010

[3] F Bashirov Spectroscopic Techniques and Hindered MolecularMotion CRC Press New York NY USA 2012

[4] P Debye Polar Molecules Reinhold New York NY USA 1929[5] N Bloembergen E M Purcell and R V Pound ldquoRelaxation

effects in nuclear magnetic resonance absorptionrdquo PhysicalReview vol 73 no 7 pp 679ndash712 1948

[6] K A Valiev and E N Ivanov ldquoRotational Brownian motionrdquoUspekhi Fizicheskikh Nauk vol 109 pp 31ndash64 1973 EnglishTranslation Physics-Uspekhi vol 16 pp 1ndash16 1973

[7] G Williams ldquoTime-correlation functions and molecularmotionrdquo Chemical Society Reviews vol 7 no 1 pp 89ndash131 1978

[8] N Bloembergen ldquoSpin relaxation processes in a two-protonsystemrdquo Physical Review vol 104 pp 1542ndash1547 1956

[9] R I Hilt and P S Hubbard ldquoNuclear magnetic relaxationof three spin system undergoing hindered rotationsrdquo PhysicalReview A vol 134 pp 392ndash398 1964

[10] P Rigny ldquoReorientations dans les cristaux moleculaires etfonctions de correlationrdquo Physica vol 59 no 4 pp 707ndash7211972

[11] R Bersohn and H S Gutowsky ldquoProtonmagnetic resonance inan ammonium chloride single crystalrdquoThe Journal of ChemicalPhysics vol 22 no 4 pp 651ndash658 1954

[12] F I Bashirov ldquoProton spin-lattice relaxation inmonocrystallineammonium chloriderdquo Journal ofMagnetic Resonance A vol 122no 1 pp 1ndash8 1996

[13] J G Powles and H S Gutowsky ldquoProton magnetic resonanceof the CH

3group III Reorientation mechanism in solidsrdquoThe

Journal of Chemical Physics vol 23 no 9 pp 1692ndash1699 1955[14] R Goc ldquoSimulation of the NMR second moment as a function

of temperature in the presence of molecular motion Applica-tion to (CH

3)3NBH3rdquo Zeitschrift fur Naturforschung vol 57 no

1-2 pp 29ndash35 2002[15] A Abragam The Principles of Nuclear Magnetism Oxford

University Press Oxford UK 1961[16] A Losche Kerninduktion Veb Deutsher Verlag der Vis-

senschaften Berlin Germany 1962[17] Ch P Slichter Principles of Magnetic Resonance Harper and

Row New York NY USA 1965[18] H S Gutowsky G E Pake and R Bersohn ldquoStructural

investigations bymeans of nuclear magnetism III Ammonium

halidesrdquoThe Journal of Chemical Physics vol 22 no 4 pp 643ndash650 1954

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


broadened owing to their magnetic dipole-dipole interactionwith resonant andwith nonresonant nuclei in a crystal latticecan be calculated by using a well-known expression [1]

119872(119894)

2= sum

119895

119872(119895119894)

2= sum

119895

3

41205742

119895ℏ2

119868119895(119868119895+ 1)

(3cos2120579119895119894

minus 1)2

1199036

119895119894

(1)

where the index 119894 labels a target resonant nucleus of a chosenmolecule and the index j other nuclei of the substance 120574

119895and

119868119895are respectively the gyromagnetic ratio and the quantum

number of the nuclei ℏ is Planckrsquos constant divided by 2120587and 120579

119895119894is the polar angle formed by the internuclear vector

r119895119894with the induction vector of the stationary magnetic field

B0determined in the laboratory reference frame (LRF)For convenience we shall replace (3cos2120579

119895119894minus 1)2

by (161205875)|119884(2)

0(120579119895119894)|2

= (161205875)|119884(2)

0(120593119895119894 120579119895119894)|2 where

119884(2)

0(120579119895119894) = 119884

(2)

0(120593119895119894 120579119895119894) is the zeroth element of normalized

spherical harmonic of the second order the zerothcomponent of the unit spherical tensor of the 2nd rankwhich depends on the polar angle 120579

119895119894 but it has no

dependence of the azimuth angle 120593119895119894 Moreover we shall

rewrite (1) in terms of spherical tensor components as

119872(119895119894)

2= sum

119895

1198722119895119894

= sum

119895

12120587

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

10038161003816100381610038161003816119884(2)

0(120593119895119894 120579119895119894)10038161003816100381610038161003816

2

(2)

By the way the angles 120579119895119894and120593119895119894relate LRFwhereas the inter-

nuclear vectors are fixed in the crystallographic referenceframe (CRF) Taking into account the transformation rule ofspherical tensors by a rotation to the three-dimensional Eulerangles (120601 120599 120585) from CRF to LRF given by the expression

119884(2)

0(120593 120579) =

2

sum

119898=minus2

119863(2)

0119898(120601 120599 120585) 119884

(2)

119898(1205931015840

1205791015840

) (3)

where 119863(2)

0119898(120601 120599 120585) is an element of Wigner matrix we can

express 119872(119895119894)

2in the form

119872(119895119894)

2= sum

119895

12120587

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

2

sum

119898=minus2

119863(2)

0119898(120601 120599 120585) 119884

(2)

119898(120593119895119894 120579119895119894)

1003816100381610038161003816100381610038161003816100381610038161003816

2

(4)

Replacing 119863(2)

0119898(120601 120599 120585) = radic(41205875)119884

(2)

119898(120599 120601) allows us to

reduce (4) to

119872(119895119894)

2= sum

119895

48120587

251205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

2

sum

119898=minus2

119884(2)

119898(120601 120599) 119884

(2)

119898(120593119895119894 120579119895119894)

1003816100381610038161003816100381610038161003816100381610038161003816

2

(5)

where (120601 120599) and (120593119895119894 120579119895119894) are respectively the spherical

angles of the vector B0and the internuclear vector r

119895119894

determined with respect to CRFAt sufficiently low temperatures a regime of rigid lattice

is valid in molecular crystals At high temperatures (T gt

40K) the direction of some internuclear vectors changesaccidentally due to classical rotational displacement Conse-quently angles (120593

119895119894 120579119895119894) become random functions of time

Such motion changes a spectral distribution law of NMRsignals Therefore the simulation of a second moment formolecular substances has to be performed by accounting forintramolecular nuclei as well as intermolecular ones

119872(119895119894)

2= 119872(intra119895119894)2

+ 119872(inter119895119894)2

(6)

The term 119872(inter119895119894)2

denoted by the label ldquo(inter119895119894)rdquo is theintermolecular part of 119872

(119895119894)

2 Its calculation is out of the

interest of this paperThe other term of the sum (6) 119872

(intra119895119894)2

labeled byldquointra119895119894rdquo is the part of 119872

(119895119894)

2delivered by nuclei inner with

respect to the given molecule It is convenient to present119872(intra119895119894)2

in the form [5 13 14]

119872(intra119895119894)2

=

119873minus1

sum

119895=1

3

41205742

119895ℏ2

119868119895(119868119895+ 1)int

120575]

minus120575]119869(0)

119895119894(120596) 119889] (7)

where 119873 is the number of identical resonant nuclear spins inthe considered molecule and

119869(0)

119895119894(120596) = int

+infin

minusinfin

119870(0)

119895119894(119905) exp (119894120596119905) 119889119905 (8)

is the nonnormalized spectral density function of the auto-correlation function

119870(0)

119895119894(119905) = ⟨119865

(0)

119895119894(119905) 119865(0)lowast

119895119894(119905 + 120591)⟩ (9)

In its turn the function 119865(0)

119895119894(119905) is the coordinate part of spin-

spin interaction Hamiltonian of nuclei 119894 and 119895 given by

119865(0)

119895119894(119905) = 119903

minus3

119895119894[3cos2120579

119895119894(119905) minus 1]

or 119865(0)

119895119894(119905) = (

16120587

5)

12

119903minus3

119895119894119884(2)

0[120579119895119894

(119905)]

(10)

The integration is prescribed within the limits of the doubleline width from minus120575] up to +120575] in (7) By the way 120575] =1T2 where T

2is the spin-spin relaxation time 120596 = 2120587] =

1205741198941198610is the resonance angular frequency of spins 119894 120574

119894is their

gyromagnetic ratio and 1198610is the module of induction vector

of the static magnetic fieldThe outcome of calculus by using (7)ndash(10) depends on

the physical model of molecular motion According to thegoal of the present study we shall apply EAJM approach[2 3] as the model of HMM within the framework of whichan analytical expression of the normalized spectral density




tensor 119884(2)


119869(2)

0(120596) = 119869

(2)

0(119902120572 120601 120599 120596)

=5

2120587sum

120572

2

sum

119897=0

119902120572120591120572

1 + (120596120591120572)21198861205721198970

(120601) cos2119897120599

(11)






the form

119869(119895119894)

0(120596) = 119869

(119895119894)

0(119902120572 120601 120599 120596) =

16120587

5119903minus6

119895119894119869(2)

0(119902120572 120601 120599 120596)

= 8119903minus6

119895119894sum

120572

2

sum

119897=0

119902120572120591120572

1 + (120596120591120572)21198861205721198970

(120601) cos2119897120599

(12)








120591120572


120572119864sum

119894

119901119894120594120572119894)

minus1

120591 (13)

where 120594120572119894

and 120594120572119864





120591 = 1205910exp(

119864119886

119877119879) (14)




119872(intra119895119894)2

=

119873minus1

sum

119895=1

61205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot int

120575]

minus120575]sum

120572

2

sum

119897=0

119902120572120591120572

1 + (2120587]120591120572)21198861205721198970

(120601) cos2119897120599 119889]

(15)



119872(intra119895119894)2

=

119873minus1

sum

119895=1

6

1205871205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot sum

120572

2

sum

119897=0

1199021205721198861205721198970


(16)


119872(intra119895119894pow)2

= sum

119895

6

51205871205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)sum

120572

119902120572arctg (2120587 sdot 120575] sdot 120591

120572)

(17)










120572119902120572


119872(intra119895119894pow)2

= sum

119895

3

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1) (18)


119872(intra119895119894)2

= sum

119895

31205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599

(19)



119872(intra1198951015840119894)2

= sum

1198951015840

4

31205742

1198951015840ℏ2

119903minus6

11989510158401198941198681198951015840 (1198681198951015840 + 1)

sdot sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599

(20)





60

50

40

300

12 2

46

3 0120599 (rad) 120601 (rad)

M(in

tra)

2G

s2



1199022= 0757 0 le 120601 le 2120587 and 0 le 120599 le 120587)










4Cl


4Cl To


4


4


4Cl we have to deal




By using 119868119895


119895= 120574H and

60

50

40

30

B 0[100]

B 0[111]

B 0[110]

3210120599 (rad)

M(in

tra)

2G

s2



[1 0 0] = 299Gs2 (120601 = 1205874 120599 = 0) and119872(intraexp)2



119903119895119894


as

119872(intraHH)2

=27

41205742

Hℏ2

119903minus6

(HH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599)

(21)

Substituting 1198681198951015840 = 1 120574

1198951015840 = 120574N and 119903

1198951015840119894= 119903NH in (20) follows


119872(intraNH)2

=8

31205742

Nℏ2

119903minus6

(NH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599) (22)





(120601) 119886100

(120601)

= (18)(1 + 3cos22120601) 119886110

(120601) = minus(34)(1 + cos22120601) 119886120

(120601)

= (38)(3 + cos22120601) 119886200

(120601) = (14)(1 minus cos22120601) 119886210

(120601) =(12)(1 + cos22120601) and 119886

200(120601) = minus(14)(3 + cos22120601) [2 3] in



as

119872(intraHH)2

(1199021 1199022 120599 120601) = 283107 sdot Ω (119902

1 1199022 120599 120601) (23)

119872(intraNH)2

(1199021 1199022 120599 120601) = 11074 sdot Ω (119902

1 1199022 120599 120601) (24)


1


Ω(1199021 1199022 120599 120601) = 119902

1[(1 + 3cos22120601)



50

40

300 1 2 2

46

3 0120599 (rad)

120601 (rad)0 1 2 2

46

3 0120599 (rad)

120601 (rad)

20

15

M(in

traH

H)

2G

s2

M(in

traN

H)

2 G

s2




4


1199022= 0757

+ 21199022[(1 minus cos22120601) + 2 (1 + cos22120601) cos2120599

minus (3 + cos22120601) cos4120599]

(25)




1 1199022 120601 and 120599

119872(intra)2

(1199021 1199022 120599 120601) = 294181 sdot Ω (119902

1 1199022 120599 120601) (26)




2


4Cl (B

0 [1 0 0]) equal to 365 Gs2



(intraexp)2

equal to 299Gs2




= 299Gs2 120601



weights 1199021and 1199022 1199021= 0254 and 119902



1= 025 and







4

+



1= 0254 and 119902

2=







B0 [1 0 0] B

0 [1 1 1] and B

0 [1 1 0]


1



119872(intraNH)2

=28310711074 = 25565

4 Conclusion






References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




tensor 119884(2)


119869(2)

0(120596) = 119869

(2)

0(119902120572 120601 120599 120596)

=5

2120587sum

120572

2

sum

119897=0

119902120572120591120572

1 + (120596120591120572)21198861205721198970

(120601) cos2119897120599

(11)






the form

119869(119895119894)

0(120596) = 119869

(119895119894)

0(119902120572 120601 120599 120596) =

16120587

5119903minus6

119895119894119869(2)

0(119902120572 120601 120599 120596)

= 8119903minus6

119895119894sum

120572

2

sum

119897=0

119902120572120591120572

1 + (120596120591120572)21198861205721198970

(120601) cos2119897120599

(12)








120591120572


120572119864sum

119894

119901119894120594120572119894)

minus1

120591 (13)

where 120594120572119894

and 120594120572119864





120591 = 1205910exp(

119864119886

119877119879) (14)




119872(intra119895119894)2

=

119873minus1

sum

119895=1

61205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot int

120575]

minus120575]sum

120572

2

sum

119897=0

119902120572120591120572

1 + (2120587]120591120572)21198861205721198970

(120601) cos2119897120599 119889]

(15)



119872(intra119895119894)2

=

119873minus1

sum

119895=1

6

1205871205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)

sdot sum

120572

2

sum

119897=0

1199021205721198861205721198970


(16)


119872(intra119895119894pow)2

= sum

119895

6

51205871205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)sum

120572

119902120572arctg (2120587 sdot 120575] sdot 120591

120572)

(17)










120572119902120572


119872(intra119895119894pow)2

= sum

119895

3

51205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1) (18)


119872(intra119895119894)2

= sum

119895

31205742

119895ℏ2

119903minus6

119895119894119868119895(119868119895+ 1)sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599

(19)



119872(intra1198951015840119894)2

= sum

1198951015840

4

31205742

1198951015840ℏ2

119903minus6

11989510158401198941198681198951015840 (1198681198951015840 + 1)

sdot sum

120572

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599

(20)





60

50

40

300

12 2

46

3 0120599 (rad) 120601 (rad)

M(in

tra)

2G

s2



1199022= 0757 0 le 120601 le 2120587 and 0 le 120599 le 120587)










4Cl


4Cl To


4


4


4Cl we have to deal




By using 119868119895


119895= 120574H and

60

50

40

30

B 0[100]

B 0[111]

B 0[110]

3210120599 (rad)

M(in

tra)

2G

s2



[1 0 0] = 299Gs2 (120601 = 1205874 120599 = 0) and119872(intraexp)2



119903119895119894


as

119872(intraHH)2

=27

41205742

Hℏ2

119903minus6

(HH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599)

(21)

Substituting 1198681198951015840 = 1 120574

1198951015840 = 120574N and 119903

1198951015840119894= 119903NH in (20) follows


119872(intraNH)2

=8

31205742

Nℏ2

119903minus6

(NH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599) (22)





(120601) 119886100

(120601)

= (18)(1 + 3cos22120601) 119886110

(120601) = minus(34)(1 + cos22120601) 119886120

(120601)

= (38)(3 + cos22120601) 119886200

(120601) = (14)(1 minus cos22120601) 119886210

(120601) =(12)(1 + cos22120601) and 119886

200(120601) = minus(14)(3 + cos22120601) [2 3] in



as

119872(intraHH)2

(1199021 1199022 120599 120601) = 283107 sdot Ω (119902

1 1199022 120599 120601) (23)

119872(intraNH)2

(1199021 1199022 120599 120601) = 11074 sdot Ω (119902

1 1199022 120599 120601) (24)


1


Ω(1199021 1199022 120599 120601) = 119902

1[(1 + 3cos22120601)



50

40

300 1 2 2

46

3 0120599 (rad)

120601 (rad)0 1 2 2

46

3 0120599 (rad)

120601 (rad)

20

15

M(in

traH

H)

2G

s2

M(in

traN

H)

2 G

s2




4


1199022= 0757

+ 21199022[(1 minus cos22120601) + 2 (1 + cos22120601) cos2120599

minus (3 + cos22120601) cos4120599]

(25)




1 1199022 120601 and 120599

119872(intra)2

(1199021 1199022 120599 120601) = 294181 sdot Ω (119902

1 1199022 120599 120601) (26)




2


4Cl (B

0 [1 0 0]) equal to 365 Gs2



(intraexp)2

equal to 299Gs2




= 299Gs2 120601



weights 1199021and 1199022 1199021= 0254 and 119902



1= 025 and







4

+



1= 0254 and 119902

2=







B0 [1 0 0] B

0 [1 1 1] and B

0 [1 1 0]


1



119872(intraNH)2

=28310711074 = 25565

4 Conclusion






References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


60

50

40

300

12 2

46

3 0120599 (rad) 120601 (rad)

M(in

tra)

2G

s2



1199022= 0757 0 le 120601 le 2120587 and 0 le 120599 le 120587)










4Cl


4Cl To


4


4


4Cl we have to deal




By using 119868119895


119895= 120574H and

60

50

40

30

B 0[100]

B 0[111]

B 0[110]

3210120599 (rad)

M(in

tra)

2G

s2



[1 0 0] = 299Gs2 (120601 = 1205874 120599 = 0) and119872(intraexp)2



119903119895119894


as

119872(intraHH)2

=27

41205742

Hℏ2

119903minus6

(HH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599)

(21)

Substituting 1198681198951015840 = 1 120574

1198951015840 = 120574N and 119903

1198951015840119894= 119903NH in (20) follows


119872(intraNH)2

=8

31205742

Nℏ2

119903minus6

(NH) sum120572=12

(

2

sum

119897=0

1199021205721198861205721198970

(120601) cos2119897120599) (22)





(120601) 119886100

(120601)

= (18)(1 + 3cos22120601) 119886110

(120601) = minus(34)(1 + cos22120601) 119886120

(120601)

= (38)(3 + cos22120601) 119886200

(120601) = (14)(1 minus cos22120601) 119886210

(120601) =(12)(1 + cos22120601) and 119886

200(120601) = minus(14)(3 + cos22120601) [2 3] in



as

119872(intraHH)2

(1199021 1199022 120599 120601) = 283107 sdot Ω (119902

1 1199022 120599 120601) (23)

119872(intraNH)2

(1199021 1199022 120599 120601) = 11074 sdot Ω (119902

1 1199022 120599 120601) (24)


1


Ω(1199021 1199022 120599 120601) = 119902

1[(1 + 3cos22120601)



50

40

300 1 2 2

46

3 0120599 (rad)

120601 (rad)0 1 2 2

46

3 0120599 (rad)

120601 (rad)

20

15

M(in

traH

H)

2G

s2

M(in

traN

H)

2 G

s2




4


1199022= 0757

+ 21199022[(1 minus cos22120601) + 2 (1 + cos22120601) cos2120599

minus (3 + cos22120601) cos4120599]

(25)




1 1199022 120601 and 120599

119872(intra)2

(1199021 1199022 120599 120601) = 294181 sdot Ω (119902

1 1199022 120599 120601) (26)




2


4Cl (B

0 [1 0 0]) equal to 365 Gs2



(intraexp)2

equal to 299Gs2




= 299Gs2 120601



weights 1199021and 1199022 1199021= 0254 and 119902



1= 025 and







4

+



1= 0254 and 119902

2=







B0 [1 0 0] B

0 [1 1 1] and B

0 [1 1 0]


1



119872(intraNH)2

=28310711074 = 25565

4 Conclusion






References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


50

40

300 1 2 2

46

3 0120599 (rad)

120601 (rad)0 1 2 2

46

3 0120599 (rad)

120601 (rad)

20

15

M(in

traH

H)

2G

s2

M(in

traN

H)

2 G

s2




4


1199022= 0757

+ 21199022[(1 minus cos22120601) + 2 (1 + cos22120601) cos2120599

minus (3 + cos22120601) cos4120599]

(25)




1 1199022 120601 and 120599

119872(intra)2

(1199021 1199022 120599 120601) = 294181 sdot Ω (119902

1 1199022 120599 120601) (26)




2


4Cl (B

0 [1 0 0]) equal to 365 Gs2



(intraexp)2

equal to 299Gs2




= 299Gs2 120601



weights 1199021and 1199022 1199021= 0254 and 119902



1= 025 and







4

+



1= 0254 and 119902

2=







B0 [1 0 0] B

0 [1 1 1] and B

0 [1 1 0]


1



119872(intraNH)2

=28310711074 = 25565

4 Conclusion






References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





References

































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

Documents

Research Article Using the Intramolecular Contribution to ...downloads.hindawi.com › archive › 2015 › 701386.pdf · Symmetry Breakdown in Molecular Crystals ... the dynamic