properties of solvated molecules

properties of solvated molecules

Kurt V. Mikkelsen

Department of Chemistry, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark Poul Jørgensen

Department of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark Kenneth Ruud and Trygve Helgaker

Department of Chemistry, University of Oslo, N-0315 Oslo, Norway

~Received 14 December 1995; accepted 8 October 1996!

We present the first model for calculating gauge-origin independent magnetic properties of solvated molecules using London atomic orbitals. The solvent is represented as a dielectric medium. We use London atomic orbitals in order to ensure gauge-origin independence and fast basis-set convergence. We present results for the magnetizability and the nuclear shielding constants of the two molecules H₂O and CH₄. © 1997 American Institute of Physics. @S0021-9606~97!50203-0#

I. INTRODUCTION

In this paper we present a rigorous method for obtaining the solvent effects on the magnetizability and nuclear shield- ing constants of solvated molecules. The method is a combi- nation of previous work on electronic Hamiltonians for origin-independent studies of magnetic properties^1–4 and on dielectric-medium response methods for calculating molecu- lar properties of solvated molecules.^5–9

The solvent is represented as a homogeneous, isotropic, and linear dielectric medium and the solvated molecule is contained within a spherical cavity embedded in the medium.^5–9The charge distribution of the solvated molecule induces a reaction field and potential in the dielectric me- dium which in turn acts on the molecular charge distribution, giving rise to a polarization energy. We perform a multipole expansion of the charge distribution of the solvated molecule and from an integral equation establish the polarization field, leading to an expression for the polarization energy. The energy functional for the solvated molecule, described by a multiconfigurational self-consistent field ~MCSCF! elec- tronic wave function, is^7–9

E~B,m!5E_vac~B,m!1E_sol~B!, ~1! where E_vac~B,m!and E_sol~B!are the energies corresponding to a Hamiltonian operator for the vacuum state and for the solvated molecular state, respectively. We note that the en- ergy functional of the vacuum state has both an explicit de- pendence on the external magnetic field through the Hamil- tonian operator and an implicit dependence through the use of magnetic field dependent orbitals ~so called London atomic orbitals¹⁰!, whereas the solvent energy has only an implicit dependence through the London orbitals. Thus, only the vacuum energy functional depends explicitly on the mag- netic moments m of the nuclei.

The magnetic properties we consider here are the mag- netizability j and the nuclear magnetic shielding constants s(K); defined respectively as^2–4

j52]^2E~B!

]^2B

U

_B50^{, ~2!}

and

s~K!511]^2E~B,m! ]^B]^mK

U

_B5m

K50

, ~3!

where m_K is the nuclear magnetic moment of nucleus K.

A phenomenological model by Buckingham has classi- fied how the solvent affects the nuclear shielding constants of a solvated molecule. The solvent contributions are written as¹¹

ssolvent5sb1sa1sw1sE, ~4!

where ssolvent is the shift of the nuclear shielding constant due to the presence of the solvent. The first term sb arises from the magnetic susceptibility of the solvent, the second term sa from the anisotropy of the magnetizability of the solvent molecules in the solvation shells surrounding the sol- ute. The third term sw relates to the van der Waal interac- tions between the solvent molecules and the solute. The last term sE accounts for the electrostatic interactions between the solute and the solvent. In what follows we will focus on the latter term sE.

II. METHOD

We begin by presenting the electronic energy functional depending explicitly on the external magnetic induction B and the magnetic moments of the nuclei m_K through the vector potential A. We then consider the solvent integrals for the London atomic orbitals¹⁰ and their derivatives with re- spect to the external magnetic field.

A. Electronic energy functional In Eq.~1!we have

E_vac~B,m!5^^O~B!uH~m,B!uO~B!&^{, ~5!}

where we have indicated all explicit dependencies on nuclear magnetic moments and external magnetic fields. In an ortho- normal basis, the spin-independent Born–Oppenheimer elec- tronic Hamiltonian is written as

H5

(

_r,s ^hrsErs112 _rs,tu

(

^{~rsutu!ers,tu, ~6!}

where we have introduced the spin-free operators

E_rs5

(

_s ^{ar†sass, ~7!}

e_rs,tu5E_rsE_tu2dstE_ru. ~8!

Here we sum over the spin quantum numbers^{and a}†ps

and a_ps are the standard creation and annihilation operators for an electron in spin-orbital fps. The one- and two-electron integrals are denoted h_rs and (rsutu), respectively. The vec- tor potential enters the one-electron Hamiltonian

h~r;A!51 2 pi

22

(

_K r^Z_Ki^K, ~9!

where we sum over all nuclei specified by the position vector R_K and the charge Z_K. The distance between the electron and the nucleus is

r_Ki5ur_i2R_Ku, ~10!

r_ibeing the position of the ith electron. In the presence of a magnetic field, the mechanical momentum is given by

pi52i¹i1A~r_i!, ~11! where the vector potential is given by

A~r!51

2 B3r_O1a²

(

_K ^mKr³_K^3rK. ~12! a denotes here the fine structure constant. The first term in Eq. ~12!represents the external magnetic field, and the sec- ond term the field originating from the nuclear magnetic mo- ments. Note that an arbitrary gauge origin O appears in the vector potential for the external magnetic field. As this origin has no physical significance, our calculated properties must be independent of this choice. This requirement is fulfilled if a complete one-electron basis is used ~at the Hartree–Fock level! or by choosing suitable local gauge origins by apply- ing magnetic-field dependent orbitals, either on the atomic^2,10,12 or the molecular^13,14 level. The use of field- dependent molecular orbitals ensures that the molecular in- tegrals are independent of the global gauge origin O. The proper choice of a connection matrix ensures orthonormality and smooth variation with respect to the magnetic field.¹⁵We shall return to the field-dependent orbitals in the next section.

The MCSCF wave function is parametrized as

uC&5exp@2ik#exp@2iS#uO&^, ~13!

where

uO&5

(

_i ^{CiuQi&. ~14!}

uQi& are configuration state functions~CSFs!, each CSF be-

ing a fixed linear combination of Slater determinants. The operator exp[2ik] describes a unitary transformation in or- bital space and is given by

k5

(

_rs ^{@krsar†as1ksr*as†ar#. ~15!}

The operator exp[2iS] describes a unitary transformation in configuration space and is given by

S5

(

n @S_nR_n^†1S_n*R_n#, ~16! where

R_n5un&^^Ou, ~17!

and the set $^un&% spans the orthogonal complement touO&^. The energy functional depends on the variational parameters of the orbital and configuration spaces and also the magnetic field B and the nuclear magnetic moments m_K. In the fol- lowing we shall refer to the variational parameters collec- tively asl, and the external perturbations~the magnetic field and the nuclear magnetic moments!as a and b.

The expressions for the magnetic parameters determined from an MCSCF state of a molecule in vacuum have been presented elsewhere.^2–4 A similar approach will be utilized here to take into account effects arising from the surrounding solvent by considering the energy functional in Eq. ~1!. We shall here focus on the second term in Eq.~1!, E_sol.

The solvent model we employ here will include only equilibrium solvent responses, which is appropriate for a ho- mogeneous magnetic field. Thus we will not consider sudden changes in the charge distribution of the solute as in the case of external electric fields of high frequency, nor will we con- sider nonequilibrium solvent states. We are therefore able to utilize the simpler equilibrium solvent energy expression with static dielectric constants; the solvent contribution to the energy can then be written as^{5–9,16–24}

E_sol~B!5

(

_lm ^{gl~^Tlm&!2. ~18!}

The function g_l depends on the shape and size of the cavity, the dielectric constant of the medium and the order of the multipole-moment expansion parameter. For a spherical cav- ity of radius R_cavembedded in a medium with dielectric con- stant e^{, we have}

g_l521

2 R_cav^2~2l11! ~l11!~e21!

l1~l11!e ^{. ~19!} The charge distribution of the solute has been expanded in a multipole series. The charge moments^^Tlm(r⁾&are expecta- tion values of the nuclear and electronic solvent operators

^^Tlm&5T_lmⁿ 2^^Tlm

e &^, ~20!

T_lmⁿ 5

(

_K ^{ZKRlm~RK!, ~21!}

T_lm^e 5R^lm~r!5

(

_pq ^{RpqlmEpq. ~22!}

The following notation is used

^^Tlm

e &⁵^^CuTlm

e uC&⁵

(

_pq ^{DpqRpqlm, ~23!}

whereuC& is the reference electronic wave function for our system@see Eq.~13!#. We have also introduced

R_pq^lm5^fpuR^lmufq&^{, ~24!}

and D_pq is the one-electron density matrix.

The functions R^lm~R_K! and R^lm are related to the con- ventional solid harmonics S_l^m,^5,9

R^l05S_l⁰, R^lm5 1

A

^{2 ~Slm1Sl2m!, ~25!}

R^l2m5 1

i

A

^{2 ~Slm2Sl2m!,}

wherem is a positive integer. The first-order partial deriva- tives of the energy functional in terms of wave function pa- rameters (li) and magnetic parameters, a and b, are given by

]^E ]li

5]^Evac

]li

1]^Esol

]li

, ~26!

]^E ]^a5

]^Evac

]^{a 1} ]^Esol

]^{a . ~27!}

The partial derivatives of the vacuum energy are given in Refs. 2, 3, and 4 and are not considered further here. The gradient of the solvent term with respect to variation of wave function parameters is given by

]^Esol

]li 522

(

_lm ^{gl^Tlm& ]^}_]^T_l^lme_i^&

522

(

lm

g_l^^Tlm&

(

pq

R_pq^lm ]^Dpq

]li

5

(

pq

t_pq ]^Dpq

]li

,

~28! where we have introduced an effective one-electron operator

t_pq522

(

_lm ^{gl^Tlm&Rlmpq. ~29!}

The gradient with respect to variations of the magnetic pa- rameters is given by

]^Esol

]^{a 522}

(

_lm ^{gl^Tlm& ]^}_]^Ta^{lme &}

522

(

lm

g_l^^Tlm&

(

pq

D_pq ]^Rpq lm

]^{a 5}

(

pq

t_pq^a D_pq,

~30!

where we have introduced the effective one-electron operator

t_pq^a 522

(

_lm ^{gl^Tlm& ]}_]^Ra^pqlm. ~31!

The response of the electronic wave function to the variation of the magnetic parameters may be expressed as

]^2Esol

]li]^a52

(

_lm ^{gl^Tlm& ]}_]²_l^{^T}_i]^lma^&12

(

_lm ^{gl]^}_]^T_l^lm_i^{& ]^}_]^Ta^lm&

5t_pq^a ]^Dpq

]li

. ~32!

We note that ]^^Tlm&^/]a is zero since the density matrix is symmetric, whereas the integrals differentiated with respect to a are antisymmetric. The Hessian with respect to the mag- netic parameters is given by

d²E_sol

dadb5]^2Esol

]^a]^{b 1} ]^2Esol

]^a]li

]l

]^{b, ~33!}

where the last term contains the wave function response to the magnetic parameters Eq. ~32!. The first term leads to

]^2Esol

]^a]^{b 52}

(

_lm ^{gl^Tlm& ]2}_]^{^}a^T]^lmb&12

(

_lm ^{gl]^}_]^Ta^lm&

]^^Tlm&

]^b 5D_pqt_pq^ab, ~34! where we have introduced the effective one-electron operator

t_pq^ab52

(

lm

g_l^^Tlm& ^]

2R_pq^lm

]^a]^{b . ~35!}

B. Field-dependent orbitals

The London atomic orbital~LAO!¹⁰centered on nucleus N is given by

v_m~R_N;A^e!5exp~2iA^e–r!x_m~R_N!, ~36! where x_m ~R_N! is an ordinary Gaussian orbital at position R_N, and exp~2iA^e–r!is the London phase factor transform- ing the global gauge origin into a local gauge origin on the nucleus to which the orbital is attached

A^e5¹2B3R_NO. ~37! Integrals involving London orbitals are gauge-origin inde- pendent as has been shown by Helgaker and Jørgensen.¹

The field-dependent orthonormalized molecular orbitals

~OMOs!are constructed as follows:

cm

OMO~r;A^e!5

(

_n ^{cnUMO~r;Ae!Tnm~B!. ~38!}

The unmodified molecular orbitals ~UMOs! are linear com- binations of London orbitals and correspond to the optimized orbitals at B50 and m50

cm

UMO~r;A^e!5

(

_m ^{Cmmvm~r;Ae!, ~39!}

where C_mmare the unperturbed MO coefficients and are in- dependent of B and m_K. The OMOs depend on the vector potential through the London orbitals and the connection ma- trix T which is chosen to be

T5W²¹~WS²¹W^†!^1/2, ~40! where S is the overlap matrix of the UMOs

S_mn5

(

_mn ^{CmmCnn^vmuvn&, ~41!}

and the matrix W is given by

W_mn5

(

_mn ^{CmmCnn^xmuvn&. ~42!}

The above matrix connection, set forth by Olsen et al. as the

‘‘natural connection,’’¹⁵ ensures minimal change in the OMOs as the magnetic field is turned on. The derivatives of the vacuum OMO integrals have been presented elsewhere;² we will concentrate here on the OMO integrals related to the interaction between the dielectric medium and the charge distribution of the solvated molecule.

The integrals of the London orbitals in Eqs. ~24! and

~25! are obtained using a transformation from integrals of Cartesian moments. The basic Cartesian solvent integralv_mn^{e f g} of the London orbitals is given by

v_mn^{e f g}5^vmux_C^ey_C^fz_C^guv_n&

5^xmuexp~iA_M!x_C^ey_C^fz_C^g exp~2iA_N!ux_n&

5^xmuexp~iA_{M N}!x_C^ey_C^fz_C^gux_n&^{, ~43!}

where e, f , and g are the orders of the Cartesian moments and

A_{M N}5¹2B3~R_{M O}2R_NO!5¹2B3R_{M N}. ~44! The subscript C indicates that the multipole expansion is taken relative to the origin of the cavity. From the above Eq.

~44! we see that the integrals@Eq. ~43!# are independent of the global gauge origin O and may be written in the form

v_mn^{e f g}5^xmuexp~iB–Q_{M N}r!x_C^ey_C^fz_C^guxn&^{, ~45!} where the antisymmetric matrix Q_{M N}has been introduced

Q_{M N}51

2

S

^{2ZY0M NM N 2XZ0M NM N 2YXM N0M N}

D

^{. ~46!}

The integralsv_mn^{e f g} depend on the magnetic field B. The derivatives of the Cartesian-moment London atomic orbital integrals with respect to the field are

]^v_mn^{e f g} ]^B

U

B50

5i

2 Q_{M N}^xmurx_C^ey_C^fz_C^guxn& ^~47!

and ]^2vmne f g

]^2B

U

_B50^{5214 QM N^xmurr˜xCeyCfzCguxn&QM N, ~48!}

where r˜ denotes the transpose of r.

The Cartesian solvent integrals are evaluated following the McMurchie–Davidson scheme for expanding Cartesian Gaussian integrals in integrals over Hermite Gaussian func- tions. As an example, we consider the overlap distribution V^{M N} of two Gaussian orbitals fixed on nucleus M at R_M with exponent a_M and nucleus N at R_N with exponent a_N,

V^{M N}5G_{i j k}~r,a_M,R_M!G_i8j8k8~r,a_N,R_N! 5t

(

50

i1i8

u

(

50 j1j8

v

(

50 k1k8

E_t^i,i8E_u^{j, j8}E_v^k,k8Ltuv. ~49! We have written the overlap distribution as a linear combi- nation of Hermite functionsLtuv,

Ltuv5 ]^t ]^Px

t

]^u ]^Py

u

]^v ]^Pz

v exp@2~a_M1a_N!r_P²# ~50! and

P5a_MR_M1a_NR_N

a_M1a_N . ~51!

Integrals over the Hermite functions are easily evaluated and the expansion coefficients are generated recursively from the equations

E₀₀⁰ 5exp

S

^{2aaMM1aaNN Qx2}

D

^{, ~52!}

E_t^i11,j5~t11!E_t^{i, j}₁₁2 a_NQ_x

a_M1a_N E_t^{i, j}1 1

2~a_M1a_N! E_t^{i, j}₂₁,

~53! E_t^{i, j11}5~t11!E_t^{i, j}₁₁1 a_MQ_x

a_M1a_N E_t^{i, j}1 1

2~a_M1a_N! E_t^{i, j}₂₁,

~54! where Q_xis the x component of the vector Q5R_M2R_N. We obtain similar equations for y and z.

The Cartesian-multipole solvent integrals contain a prod- uct of the Gaussian overlap distribution V^{M N} and x_C^ey_C^fz_C^g allowing us to write

V^{M N}x_C^ey_C^fz_C^g

5ⁱ¹_t

(

ⁱ₅⁸¹₀^{m j1}_u

(

₅^j8₀^{1n k1}_v5

(

^k8₀^{1o eEti,i8 fEuj, j8gEvk,k8Ltuv. ~55!}

The expansion coefficients are obtained from the relation

eE_t^i,i85^e21E_t^i11,i81X_{M C}^e21E_t^i,i8, ~56! where X_{M C} is the x coordinate of nucleus M relative to the origin of the cavity. Equation~56!follows from

x_CG_i5~x_M1X_{M C}!G_i5G_i111X_{M C}G_i. ~57! The integral ^xmurx_C^ey_C^fz_C^gux_n& appearing in the expres- sion for the B derivatives of the solvent integrals Eq. ~47!,

and similarly in Eq. ~48!, contain two different origins: ~i! the origin of the Cartesian coordinate frame and ~ii!the ori- gin of the cavity, which means that our expansion coeffi- cients will differ depending on the origin. For the first de- rivative with respect to the magnetic field induction, Eq.

~47!, we obtain overlap distributions of the form

xV^{M N}x_C^ey_C^fz_C^g5 t

(

50 i1i8^1e

u

(

50 j1j8^1f

v

(

50 k1k8^1g

@^eE_t^{i11,i8 f}E_u^{j , j8 g}E_v^k,k8

1X_M^eE_t^{i,i8 f}E_u^{j , j8g}E_v^k,k8#Ltuv, ~58! where X_M is the x coordinate of nucleus M relative to the Cartesian coordinate frame origin. The second derivatives with respect to the magnetic field Eq.~48!, may be obtained from overlap distributions of, for example

xyV^{M N}x^ey^fz^g

5 t

(

50 i1i8^1e

u

(

50 j1j8^1f

v

(

50 k1k8^1g

@^eE_t^{i11,i8 f}E_u^j11,j8gE_v^k,k8

1Y_M ^eE_t^{i11,i8 f}E_u^{j , j8g}E_v^k,k8 1X_M^eE_t^{i,i8 f}E_u^{j11,j8 g}E_v^k,k8

1X_MY_M ^eE_t^i,i8fE_u^{j , j8g}E_v^k,k8#Ltuv, ~59! where Y_M is the y coordinate of nucleus M relative to the Cartesian coordinate frame origin.

III. CALCULATIONS

We have carried out calculations of magnetizabilities and nuclear shielding constants for the two molecules H₂O and CH₄ at the Hartree–Fock level using the basis-set 6-31111G(2d,2p).²⁵ As solvents we have chosen pentane

~e51.844!, benzene ~e52.28!, ethyl acetate ~e56.02!, hex- anol ~e513.3!, acetone ~e520.7!, methanol ~e532.63!, and water ~e578.54!. For each solvent the magnetic properties are calculated for fully optimized solute molecular struc- tures. The multipole expansion is truncated at l_max510 un- less otherwise stated. The cavity sizes~radii!of the two com- pounds are: 3.9 au~H₂O!and 4.13 au~CH₄!. In addition, we have considered variations of the cavity size for H₂O. The preferred cavity radius was taken to be equal to the sum of the distance from the center of mass to the most distant atom plus the van der Waal radius of that atom.

We consider in the present work the total isotropic mag- netizabilities and nuclear shieldings. In addition, we study the diamagnetic and paramagnetic contributions to these properties separately. This separation is made possible by the natural connection.¹⁵ For the magnetizability, we also con- sider the first and second anisotropies

Dj15jaa2jbb1jcc

2 , ~60!

Dj25jbb2jaa1jcc

2 , ~61!

where jaa, jbb, and jcc are the diagonal elements of the magnetizability tensor, jaa being the smallest and jbb the largest of the three components.

The values obtained for the magnetizability of the mol- ecules in vacuum are collected in Table I, and the vacuum values for the nuclear shielding constants are shown in Table II. Comparison between calculated and measured properties^2–4 shows that the vacuum calculations at the Hartree–Fock level give a good description of the magnetic properties. Therefore, the solvent effect due to the reaction field is expected to be appropriately described by an SCF reference wave function.

A. Magnetizability 1. H₂O

In Fig. 1~a! is shown the change in the magnetizability of H₂O for different choices of dielectric constants and vari- ous multipole truncations. For all dielectric constants and all orders of the multipole expansion, the solvent introduces a diamagnetic shift in the magnetizability. The solvent contri- bution to the magnetizability has converged for expansions of order l_max.3. Thus the Onsager approach, corresponding to a truncation of the multipole expansion at l_max51, is clearly inadequate.

The change in the diamagnetic contribution is shown in Fig. 1~b!. After a brief increase with respect to the dielectric constant~until aboute56!it reaches a more or less constant value. The paramagnetic contribution decreases with increas- ing dielectric constant, see Fig. 1~c!. It is noteworthy that it is mainly the change in the paramagnetic contribution that determines the overall shift of the magnetizability with the dielectric constant. As for the total magnetizability, both the diamagnetic and the paramagnetic solvent contributions con- verge for multipole expansions truncated at l_max.3.

In Figs. 1~d!and 1~e!we have plotted the first and sec- ond anisotropies respectively against the dielectric constant of the surrounding solvent for different truncations of the multipole expansion. Compared to their vacuum values, the first anisotropy increases with increasing dielectric constants, whereas the second anisotropy decreases with increasing di- electric constant. The two anisotropies thus behave quite dif- ferently upon solvation. We also note that the Onsager model overshoots the solvent effect on the first anisotropy by more than a factor of 2.

We will now consider the effect of the cavity size on the calculated magnetizability. In Fig. 2~a!we have plotted the isotropic magnetizability for different multipole expansions and various sizes of the cavity ate578.54. We note that the isotropic magnetizability depends strongly on the size of the

cavity, as can be seen from Eq. ~19!. For all cavities, the isotropic magnetizability converges for expansions with l_max.3.

Figures 2~b! and 2~c! show how the dia- and paramag- netic contributions to the isotropic magnetizability of water change with respect to the multipole expansion and changes

in the cavity size. The dependence is similar to that of the isotropic magnetizability, but more pronounced for the para- magnetic part than for the diamagnetic.

The influence of the cavity size on the anisotropies is shown in Figs. 2~d! and 2~e!. Interestingly, the first anisot- ropy varies only slightly with respect to multipole expansion

FIG. 1.~a!The isotropic magnetizability of H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion. ~b! The isotropic diamagnetic contribution to the magnetizability of H₂O vs the di- electric constant of the solvent for different truncations of the multipole expansion.~c!The isotropic paramagnetic contribution to the magnetizabil- ity of H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.~d!The first anisotropy of the magnetizability of H₂O vs the dielectric constant of the solvent for different truncations of the multipole expansion.~e! The second anisotropy of the magnetizability of H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.

TABLE I. Vacuum values of the magnetizability for the molecules investigated. All values reported are in units of 10²³⁰J T²². The corresponding experimental values are presented in parentheses.

Molecule j j^dia j^para Dj1 Dj2

H2O 2229.2@218630~Ref. 28!# 2255.2 26.0 23.3@21.760.4~Ref. 28!# 3.1@23.960.2~Ref. 28!#

CH4 2314.4@289613~Ref. 29!# 2477.7 163.3

and cavity size for the multipole expansions with l_max.1. In contrast, the second anisotropy varies as strongly as the iso- tropic magnetizability with respect to the multipole expan- sion and cavity size.

2. CH₄

Figure 3 illustrates the variation of the change of the diamagnetic, paramagnetic and total isotropic magnetizabil-

FIG. 2. ~a!The isotropic magnetizability of H₂O vs the truncation of the multipole expansion for different radii of the cavity. The solvent is water with a dielectric constant of 78.54.~b!The diamagnetic contribution to the isotropic magnetizability of H2O vs the truncation of the multipole expan- sion for different radii of the cavity. The solvent is water with a dielectric constant of 78.54.~c!The paramagnetic contribution isotropic magnetizabil- ity of H2O vs the truncation of the multipole expansion for different radii of the cavity. The solvent is water with a dielectric constant of 78.54.~d!The first anisotropy of the isotropic magnetizability of H2O vs the truncation of the multipole expansion for different radii of the cavity. The solvent is water with a dielectric constant of 78.54.~e!The second anisotropy of the isotro- pic magnetizability of H2O vs the truncation of the multipole expansion for different radii of the cavity. The solvent is water with a dielectric constant of 78.54.

TABLE II. Vacuum values of the nuclear shieldings tensors. The corresponding experimental values are presented in parentheses. All values reported in ppm.

Molecule Nucleus s s^dia s^para

H2O O 334.6@344.0617.2~Ref. 30!# 417.2 282.6

H 31.87@30.05260.015~Ref. 31!# 25.12 6.75

CH4 C 196.7@198.7~Ref. 32!# 298.1 2101.4

H 31.98@30.61~Ref. 31!# 28.97 3.01

ity versus the dielectric constant of the surrounding solvent.

We note that effects of the solvent are the same as for H₂O.

The absolute values of the diamagnetic and the paramagnetic magnetizabilities decrease for an increasing dielectric con- stant, and the paramagnetic contribution dominates the sol- vent induced change in the magnetizability.

B. Nuclear shielding constants 1. H₂O

Figures 4~a!and 4~b!show the solvent shifts~relative to the vacuum values! of the isotropic shielding constants of oxygen and hydrogen as a function of dielectric constant and truncation of the multipole expansion. The isotropic shield- ing constant for oxygen increases whereas the shielding of hydrogen decreases with respect to an increasing dielectric constant. Note also that the multipole expansion converges for l_max>3 and that the Onsager dipole model accounts for less than 50% of the solvent shift.

For oxygen, the paramagnetic contribution is more sen- sitive to the properties of the dielectric medium than the diamagnetic; compare Figs. 4~c!and 4~d!. Both the paramag- netic and diamagnetic contributions are enhanced by the me- dium. For hydrogen on the other hand, both the dia- and paramagnetic solvent shifts are diamagnetic, see Figs. 4~e! and 4~f!. Again we find that the Onsager model is inad- equate. As for oxygen, the influence of the solvent is most pronounced for the paramagnetic part of the shielding.

The solvent shifts of the isotropic shielding depend, as seen from Figs. 5~a!–5~f!, strongly on the size of the cavity, see Eq.~19!. As expected, we approach the vacuum values as the cavity radius increases. It is interesting to note that for expansions with l.2 we can express the dependency of the oxygen solvent shift on the cavity radius by the formula b 3R_cav^2a, which for l_max510 gives b52.31310³and a54.32.

As the paramagnetic term dominates the oxygen solvent shift, we may approximate the dependency of the paramag- netic term by the same expression, obtaining for l_max510, b52.91310³and a54.45. The solvent shifts of the diamag- netic part of the shielding depend on the cavity radius in a more complicated manner, which we may approximate by c1dR_cav1eR_cav² 1f R_cav³ . For l_max510 we find that c 5250.42, d534.36, e527.90, and f50.61.

The dependence of the hydrogen solvent shifts on the cavity size also follows the functional form c1dR_cav1eR_cav² 1f R_cav³ . Using a multipole expansion with l_max510, we obtain the following values for the fit of the hydrogen solvent shift: c5275.74, d549.91, e5211.22, and f50.85. Similarly, the values obtained for the dia- and paramagnetic parts are c5251.72, d534.60, e527.86m f50.60, and c5224.08, d515.36, e523.37, f50.25, re- spectively.

2. CH₄

For carbon, the solvent shift in the isotropic shielding constant increases when increasing the dielectric constant of the medium, see Fig. 6~a!. As seen from Fig. 6~a!, the changes in the dia- and paramagnetic contributions are of

opposite sign and the paramagnetic change is almost five times as large as the diamagnetic one. In Fig. 6~b!we present the solvent induced shifts for these two contributions. In contrast to the solvent shifts on the hydrogen shieldings of water, the solvent shifts of the hydrogen shieldings have op- posite signs, and we also note that the shift in the diamag- netic term is about a factor of three larger than that of the shift in the paramagnetic term.

IV. SUMMARY

The magnetizability for all compounds investigated here becomes more diamagnetic when increasing the dielectric constant of the medium~diamagnetic shift!. The solvent shift is governed primarily by the paramagnetic contribution to the magnetizability. For water, the anisotropies of the mag- netizability are strongly affected by the dielectric constant.

The solvent effect is largest for the compound with the larg- est charge moments, which is easily understood from the underlying model. We also note that the magnetizability shift depends strongly on the truncation of the multipole expan- sion of the solute. The Onsager approximation ~retaining only the dipole term!is clearly insufficient for describing the effects of the dielectric medium on the magnetizability.

With regard to the nuclear shieldings we observe that the shielding constant of the most electronegative atom in a compound increases ~diamagnetic shift! with increasing di- electric constant whereas the constants of the other atoms decrease ~paramagnetic shift!. For oxygen and carbon the paramagnetic contribution dominates the behavior of the nuclear shielding constant in a dielectric medium. The hy- drogen shieldings in the molecules studied here decrease with increasing dielectric constant and this is related to a decrease in both the paramagnetic and diamagnetic contribu- tions. For the hydrogen atoms, it is the diamagnetic contri- bution to the shielding that dominates the solvent shift.

For water, we have studied how the size of the cavity influences the magnetic properties and have seen that their dependence on the cavity size follows the simple expression

FIG. 3. The change in the isotropic magnetizability and the diamagnetic and paramagnetic contributions to the isotropic magnetizability of CH4vs the dielectric constant of the solvent.

c1dR_cav1eR_cav² 1f R_cav³ . It is by no means surprising that the calculated properties depend on the cavity size, see Eq.

~19!.

Although not given much prominence in the present in- vestigation, we would like to stress the importance of using solute optimized molecular geometries at all stages of the calculations. Often the effects due to changes in the geom-

etry oppose the effects observed from changes in the dielec- tric medium alone for a fixed vacuum geometry.²⁶ By the same token the change in the geometry alone, as employed in previous investigations of solvent shifts of magnetic properties,²⁷ will also give an unbalanced description of the effects of a dielectric medium on magnetic properties.

The present work has concentrated on one particular sol-

FIG. 4. ~a!The isotropic shielding constant for oxygen in H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.

~b!The isotropic shielding constant for hydrogen in H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.~c!The paramagnetic contribution to the isotropic shielding constant for oxygen in H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.~d!The diamagnetic contribution to the isotropic shielding constant for oxygen in H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.~e!The paramagnetic contribution to the isotropic shielding constant for hydrogen in H₂O vs the dielectric constant of the solvent for different truncations of the multipole expansion.~f!The diamagnetic contribution to the isotropic shielding constant for hydrogen in H2O vs the dielectric constant of the solvent for different truncations of the multipole expansion.

vent effect of molecular magnetic properties. Buckingham has presented a classification of the different solvent contri- butions to the nuclear shielding constants¹¹

ssolvent5sb1sa1sw1sE, ~62!

where ssolvent is the shift of the nuclear shielding constant due to the presence of the solvent. The first term sb arises

from the magnetic susceptibility of the solvent, the second term sa from the anisotropy of the magnetizability of the solvent molecules in the solvation shells surrounding the sol- ute. The third term sw relates to the van der Waal interac- tions between the solvent molecules and the solute, whereas the last term sE accounts for the electrostatic interactions between the solute and the solvent. It is thus the latter term

FIG. 5. ~a!The isotropic shielding constant for oxygen in H2O vs the radius of the cavity.~b!The isotropic shielding constant for hydrogen in H2O vs the radius of the cavity.~c!The diamagnetic contribution to the isotropic shielding constant for oxygen in H2O vs the radius of the cavity.~d!The paramagnetic contribution to the isotropic shielding constant for oxygen in H₂O vs the radius of the cavity.~e!The diamagnetic contribution to the isotropic shielding constant for hydrogen in H2O vs the radius of the cavity.~f!The paramagnetic contribution to the isotropic shielding constant for hydrogen in H2O vs the radius of the cavity.

sEthat has been the focus of this work. However, in order to reproduce experimentally observed shifts, all four terms need to be included, as shown in a recent study on the magnetic properties of fluoromethanes.²⁶

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