Static and dynamic approaches for the calculation of NMR parameters:

Permanganate ion as a case study

Ilaria Ciofini, Carlo Adamo *

Laboratoire d’Electrochimie et Chimie Analytique, CNRS UMR-7575, Ecole Nationale Superieure de Chimie de Paris,

11 rue P. et M. Curie, F-75231 Paris CEDEX 05, France

Received 21 October 2005; accepted 24 October 2005

Available online 27 January 2006

Dedicated to Annick and to her French touch to DFT

Abstract

Magnetic shielding constants of the permanganate ion ðMnO�4 Þ were computed by the means of density functional theory both from static and

dynamic simulations. The hybrid PBE0 exchange correlation functional was used for the determination of structural and magnetic properties. Ab-

initio molecular dynamic simulations were performed at the same level of theory using the atom-centred density matrix propagation (ADMP)

method. With the aim of understanding the role of coupling of different vibrations, the results obtained at static level and as average along large

amplitude motions representing each of the normal modes of the ion were compared to those resulting from dynamic approaches.

q 2005 Elsevier B.V. All rights reserved.

Keywords: ADMP; NMR chemical shieldings; Vibrational average; DFT; Permanganate ion

1. Introduction

With the recent development of efficient ab-initio molecular

dynamic approaches [1] an increasing number of effects, such as

environment or temperature, on the magnetic properties of

molecular systems can be explicitly included in the simulations

[2–4]. Nevertheless, purely quantum effects such as zero point

vibrational contributions cannot be reproduced using a classical

propagation of the nuclei, as it is the case of any (classical or

ab-initio) molecular dynamic simulations. In fact, the proper

treatment of temperature effects and zero point vibrational

contributions (ZPVC) claims for a quantum treatment of the

motion of the nuclei that is the solution of the nuclear ro-

vibrational Schrodinger equation. If, for several properties, a

static approach can be sufficient to correctly describe the system,

it has been proven that NMR properties, such as the shielding

temperature dependence or indirect spin–spin coupling, are very

sensitive to nuclear geometries and usually determined by

ro-vibrational contributions [5]. Furthermore, especially in the

case of very floppy molecules, higher vibrational states can be

populated at room temperature inducing a deformation of the

0166-1280/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2005.10.057

* Corresponding author.

E-mail address: carlo-adamo@enscp.fr (C. Adamo).

system’s structure with respect to the optimised static structure.

These structural deformations, normally referred to as large

amplitude motions (LAMs), can induce a sizable change in the

magnetic properties of the system probing its local chemical

environment. To take this effect into account one can define a

one-dimensional coordinate representing a LAM and exactly

solve the one-dimensional ro-vibrational Schrodinger equation

along that mode, averaging the properties under analysis [6,7].

This approach has been extensively applied to the calculation of

nuclear magnetic shielding of diatomic molecules [8] as well as

to the calculation of vibrationally averaged EPR properties

(A and g tensor) of larger organic radicals [9,10]. Nevertheless,

in the case of several LAMs affecting the property of interest, the

solution of an n-dimensional nuclear ro-vibrational Schrodinger

equation should be envisaged. Nowadays, this task can be

accomplished by perturbative [11,12] and variational

approaches [13] even in the case of relatively large molecular

systems [14]. These approaches are suited for semi-rigid

systems where the effect of anharmonicity on the potential

energy surface can be considered as a perturbation (up to the k

order) of the potential energy surface around the harmonic

minimum or, differently said, when the harmonic normal modes

are a good description, even beyond the harmonic region.

Inspired by a work, reporting the computed nuclear

magnetic shielding of permanganate ion in aqueous solution

both at static and at dynamic level [15], we decided to consider

this semi-rigid system as a test case to investigate the

Journal of Molecular Structure: THEOCHEM 762 (2006) 133–137

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I. Ciofini, C. Adamo / Journal of Molecular Structure: THEOCHEM 762 (2006) 133–137134

differences between static, vibrationally averaged and dynamic

approaches for the calculation of magnetic shielding in the gas

phase.

In this paper, we are not interested in reaching the chemical

accuracy in the calculation of magnetic shielding for both Mn

and O atoms nor at understanding from one side the large

deviations in manganese shielding constant with respect to the

experimental values and from the other side the role of solvent,

but only in the comparison of three above mentioned

approaches. In particular, since it has been shown that there

is a dependence of the 55Mn shielding on the manganese to

oxygen distance but also on the bending and torsion angles

[15,16], we want to investigate if the dynamic behaviour of the

ion at room (or higher) temperature can be described simply by

averaging along one-dimensional LAMs or if a full perturba-

tive treatment of the ro-vibrational problems should be

considered.

In order to evaluate possible coupling between fundamental

harmonic vibrations, a perturbative anharmonic frequency

analysis has also been performed. This analysis allowed us to

evaluate the magnitude of Coriolis couplings and the

magnitude of the coupling via cubic and quartic terms in the

potential, eventually important for higher energy modes. Our

results clearly indicate that the harmonic modes are not

significantly coupled neither by Coriolis nor by anharmonic

couplings. As a consequence, the n-dimensional ro-vibrational

problem can be treated as the sum of n-(uncoupled) one-

dimensional one. Furthermore, this treatment allows not only

the calculation of ZPVC to nuclear magnetic shieldings but

also the inclusion of temperature effects.

Our results clearly underline how, in this case, a complete

dynamic treatment is not necessary and how simplified, and

less expensive, one-dimensional approaches can be fruitfully

used. Yet, it should be kept in mind that averaging along single

LAMs is only possible if the modes are not coupled and if a

suitable vibrational coordinate is chosen, while dynamic

approaches or a full ro-vibrational treatments do not imply

any a priori knowledge on the physics of the system under

analysis.

2. Computational details

All calculations were performed using the Gaussian 03

program package [17]. The hybrid PBE0 exchange-correlation

functional [18], mixing 1/4 of Hartree Fock exact exchange to

the PBE exchange [19], was used throughout for structural and

NMR calculations both at dynamic and static level. All

structural optimisations were performed using the Los Alamos

double zeta valence basis (hereafter LANL2 [20]) and the

corresponding pseudo-potential for the Mn atom [21]. The

same basis set was used for all ADMP simulations and to

construct the one-dimensional potential energy surface along

each normal mode. A larger, all electron basis set (hereafter

AE) was used to compute magnetic shielding on stationary

point and on snapshots along the ADMP trajectories. The AE

basis consists have a [8s6p4d] [22] Watchers’ basis on Mn

(contraction scheme 62111111/331211/3111) [23] while the

oxygen atoms are described using a [3s2p1d] MIDI! basis

(contraction scheme 321/21/1) [24]. Shielding constants were

computed using the GIAO formalism.

Gas phase, unconstrained NVE ADMP simulations [25–27]

were performed starting from the equilibrium geometry giving

two different initial nuclear kinetic energies 1000 and

2850 mHartree, respectively. The fictitious electron mass has

been fixed to 0.2 amu and a time step of 0.2 fs has been used for

a global simulation time of 1.6 ps. Snapshots were taken along

the trajectories each 20 fs.

Harmonic frequencies were obtained as analytical second

derivatives while cubic and quartic force constants were

derived by finite difference of analytical second derivatives,

vibrational levels being subsequently computed by a perturba-

tive approach as described in Ref. [12] and implemented in the

Gaussian package.

Averages over large amplitude motions representing each of

the normal modes of the ion were computed solving the one-

dimensional ro-vibrational Schrodinger equation using the

program DiNa [7].

Solvent effects were included using the polarizable

continuum model (PCM) of Miertus Scrocco and Tomasi

[28] using the Cosmo PCM implementation of Cossi and

Barone [29].

3. Results and discussion

In Table 1, the computed Mn–O distance both at optimized

geometry and as average on the ADMP trajectories is reported

with respect to previous estimations from literature and to the

available experimental data. It should be noticed that contrary

to previous studies [15], in this paper, the same basis and

exchange correlation functional were used to describe the

system at static and ADMP level, thus making the direct

comparison of structural and magnetic results straightforward.

Furthermore, the same functional has been used to compute

both structural and magnetic properties. Due to the small basis

set used, the MnO distance is underestimated of about 0.03 A, a

better agreement being reachable when using a larger basis set

[15]. Inclusion of solvent at static level, slightly shortens the

bond distance (of 0.006 A). This result is in qualitative and

quantitative agreement with previous calculations explicitly

including solvent (water) at dynamic level (Car-Parrinello

simulations, CP) [15] thus underlying the validity of the PCM

model at least for structural determination. Indeed, it should be

noticed that the PCM alone is not able to reproduce the effects

due to a direct (H bond) interaction between solute and solvent.

Since it has already been shown [15] that, in the case of the

calculation of shielding for permanganate in water, the latter

effect on the property is very important, we already expect to

obtain an incorrect description when computing the shielding

using PCM (see below). On the other hand, very small

differences can be noticed between static and dynamic average

values, being of the order of 0.0002 A. A larger difference

(0.002 A) was found in literature using NVE CP simulations at

300 K and a GGA functional [15], the difference between our

Table 1

Main geometrical parameter (d(Mn–O) distance, in A), harmonic vibrational frequencies (n, in cmK1) and anharmonic corrections (Dn, in cmK1) computed for

MnOK4 in gas phase and in water solution (PCM) with respect to literature and experimental data

BP86/AE1a BP86 CP/opt

(gas phase)bBP86 CP/av

(gas phase)cBP86 CP/av

(C28 H2O)d

PBE0/LANL2

(gas phase)

PBE0/LANL2

(PCM)

PBE0/ADMP-

LANL2 (Gas

phase)d

Exp.

D(Mn–O) 1.625 1.622 1.624(30) 1.621(26) 1.599(12) 1.593 1.599(32)e 1.629G0.005g

1.599(64)f G

n2 (E) 345 369 – – 367 462 – 360h

n4 (T2) 392 424 – – 417 505 – 430

n1 (A1) 880 979 – – 936 877 – 839

n3 (T2) 949 1049 – – 1009 937 – 914

Dn2 (E) – – – – 2 – – –

Dn4 (T2) – – – – 2 – – –

Dn1 (A1) – – – – 11 – – –

Dn3 (T2) – – – – 14 – – –

a Gas phase results from Ref. [16]. The AE1 basis set consists of a Watchers’ basis on Mn and of a 6-31G* on O.b Gas phase results, from Ref. [15].c Average value from CP simulations performed in gas phase, from Ref. [15].d Running average on ADMP trajectory.e initial nuclear kinetic energy of 1000 mHartree.f initial nuclear kinetic energy of 2850 mHartree.g From Ref. [31].h From Ref. [32].

I. Ciofini, C. Adamo / Journal of Molecular Structure: THEOCHEM 762 (2006) 133–137 135

results and the one in literature being related both to the

different functional and basis set used.

The harmonic frequencies (also reported inTable 1) computed

in gas phase are in agreement with previous estimations and

experimental values, all being systematically overestimated

consistently to the shorter Mn–O computed bond length. This

effect is enhanced by the presence of solvent (PCM results in

Table 1) in the case of the n2 and n4 vibrations while the opposite

holds for the higher frequency vibrations (i.e. the stretching

modes n1 and n4) that are shifted towards smaller wave numbers,

in better agreement with the experimental data.

Anharmonic corrections computed at the same level of

theory in the gas phase are reported in Table 1. Very small

corrections are found for all modes, the largest being of

14 cmK1 for the asymmetric stretching mode (n3).

More importantly, the inspection of the computed cubic

terms of the potential and of the Coriolis couplings and

rotational constants shows that all these terms are negligible,

thus underling the absence of coupling between different

normal modes. This allows to treat the different normal modes

as uncoupled and thus to solve independent one-dimensional

ro-vibrational Schrodinger equation for each mode in order to

Table 2

Computed magnetic shielding constants (s) for MnOK4 . In the case of ADMP simulat

on snapshot taken along the trajectory

B3LYP/

BP86-AE1/II

(gas phase)a

B3LYP/BP86-

CPopt/II (gas

phase)a

B3LYP/

BP86-CPav/II

(gas phase)a

B3LYP/BP86-

CPav/II (C28

H2O)a

s(55Mn) K4832 K4794 K4829 K4830

s(17O) K1094 K1084 K1095 K1053

The following notation is used: functional for shielding calculations/functional anda From Ref. [15]. The basis set II consists have a Watchers’ basis on Mn and ofb Initial nuclear kinetic energyZ1000 mHartree.c Initial nuclear kinetic energyZ2850 mHartree.

get zero point vibrational contributions and temperature effects

on the computed 55Mn shielding. To this end, potential energy

profiles along each of the normal mode of the molecule were

computed point by point beyond the harmonic range and the

full ro-vibrational one-dimensional Schrodinger equation

along each mode has been solved. Next, the temperature

dependence of the magnetic shielding along a specific mode

was computed assuming a Boltzmann population of the

vibrational levels. The thermally averaged magnetic shielding

can thus be expressed as:

hsiT Z se C

PjhjjDsjji exp½ð30K3jÞ=kT�P

jZ0 exp½ð30K3jÞ=kT�(1)

Ds being the variation of the shielding along the selected

normal mode and j the corresponding vibrational levels.

The absolute nuclear magnetic shielding (s(55Mn) and

s(17O)) computed on the optimized structure (static level) and

as average along the ADMP trajectories are reported in

Table 2. In Fig. 1 are reported the evolution of the s(55Mn)

computed on the snapshot along the ADMP trajectories and the

corresponding running averages. The s(55Mn) computed at

PBE0/AE static level is roughly 200 ppm larger than that,

ions (column 4 and 5), the values represent an average over the values computed

PBE0/

PBE0/AE

(gas phase)

PBE0/PBE0/

AE (PCM)

PBE0/PBE0/

AE

hADMPi310 Kb

PBE0/PBE0/

AE

hADMPi900 Kc

Exp.

K4619 K4521 K4627 K4640 K

K825 K795 K827 K830 K939

method for geometry calculation/basis set for shielding calculations

contracted Huzinaga on O (Ref. [15]).

Table 3

Computed s(55Mn) for MnOK4 averaged over the large amplitude motions

(LAMs) representing the ion normal modes

s(55Mn)

E T2 A1 T2

0 K K4625 K4625 K4639 K4620

298 K K4627 K4628 K4639 K4620

900 K K4638 K4637 K4650 K4621

For comparison the static values computed at the same level of theory,

se(55Mn), is K4619 (this work Table 2).

Fig. 1. Computed s(55Mn) on selected snapshots along the ADMP trajectories. Open circles: ADMP simulation performed at 900 K; open triangles: ADMP

simulation performed at 300 K. Full circles and full triangles represent the corresponding running averages.

I. Ciofini, C. Adamo / Journal of Molecular Structure: THEOCHEM 762 (2006) 133–137136

previously reported in literature, computed at BP86/AE1 static

level [15]. This discrepancy can be fully ascribed at the

difference in basis set and functional used that is also reflected

by the difference in computed equilibrium geometries.

Comparing the shielding computed at PBE0 level in the gas

phase at static level to that obtained when averaging on

molecular dynamics snapshots, a small decrease of the s(55Mn)

is found (K8 and K31 ppm) depending on the initial nuclear

kinetic energy given while the s(17O) is practically unaffected.

On the other hand, a much larger effect, of opposite sign, is

computed when adding the solvent reaction field. In particular,

the static values for s(55Mn) and s(17O) increase of 98 and

30 ppm, respectively. Indeed, it has been shown [15] that the

indirect effect on s(55Mn), that is induced by a change in

nuclear geometry due to the presence of the solvent, is

somehow completely balanced by direct effect (H bonding with

the solvent) of opposite sign. As a consequence our simulations

in PCM are not a realistic description of the behaviour of

s(55Mn) in the aqueous medium since they do not account for

direct effects. On the other hand, the PCM model seems to

qualitatively recover the behaviour of s(17O), thus suggesting

that in this case indirect effects play the major role.

Let us now consider the average of s(55Mn) over the LAMs

(here, the normal modes) reported in Table 3. As mentioned in

the introduction a full ro-vibrational treatment allows for the

determination of ZPVC. As already found for light atoms [30]

these corrections are computed to be important also for heavier

atoms. In fact, in our case s0 KKse is computed to be of 6 ppm.

In this context, it is also important to stress that a meaningful

estimation of the ZPVC should take into account the

contribution of nuclear all degrees of freedom (here, all normal

modes) and that a correction computed only a selected degree

of freedom (that is s0 KKse along a selected normal mode for

instance the totally symmetric stretching) could lead to

misleading results.

Our findings underline that any classical propagation of the

nuclei (molecular dynamic) at very low temperature (i.e. close to

0 K) will give a wrong average value for the shielding due to the

neglecting of ZPVC. Indeed, at room temperature the average

compute along the ADMP trajectories and the vibrational

averaging (s298 K) are fully equivalent. This is clearly highlighted

by comparisonofTables 2 and 3. TheADMP trajectory evaluated

with an initial nuclear kinetic energy of 1000 mHartreecorresponds to a simulation performed atw316 K and therefore,

the average shielding computed on snapshot taken from this

trajectory (Table 2) can be compared to vibrational averaging

computed at 298 K. At this temperature, the n2 (E) and n4 (T2)

vibrational states will mostly be populated and, assuming a

Boltzmann population of the different modes, from Table 3 we

derive that the shift with respect to the static value (s298 KKse) is

of 8.6 ppm. This value is in excellent agreement with the shift

computed from ADMP results (hs310 KiKse) of 8 ppm (from

Table 2) and it underlines how temperature effects are correctly

recovered by molecular dynamic simulations at room

temperature.

Indeed, it is worthwhile to stress that a meaningful

vibrational average along the normal modes at room

temperature should not be performed only along the stretching

modes (n1 (A1) and n3 (T2)) since they will not be significantly

populated at this temperature. Furthermore, even if it is

appealing to imagine that only the stretching modes will

contribute to change the Mn–O distance, it is important to

remember that all modes implies a variation of the Mn–O

Fig. 2. Schematic representation of n2 (E symmetry, left) n4 (T2 symmetry, middle) and n3 (T2 symmetry, right) normal modes.

I. Ciofini, C. Adamo / Journal of Molecular Structure: THEOCHEM 762 (2006) 133–137 137

distance even the ones normally indicated as bending or

torsion, that is the n2 (E) and n4 (T2) modes, graphically

depicted in Fig. 2. It is also clear that in this case the stretching

contribution will be smaller and thus consistently the effect on

the computed shielding along the mode smaller.

In summary, the comparison of the chemical shift of 55Mn at

room temperature with respect to its static value (sroomKse)

computed on snapshot along the ADMP trajectory or

vibrationally averaging along single modes shows that there is

a quantitative agreement between the two approaches if all

modes thermally populated (but uncoupled) are considered. At

higher temperature (900 K) all modes will be significantly

populated and thus also the (n1 (A1) and n3 (T2)) vibrations will

contribute to the chemical shift. Boltzmann weighting allows to

compute an overall shift (hs900 KiKse) of 18 ppm to be

compared to 21 ppm computed as average on ADMP

trajectories. Once again, there is an excellent agreement

between the vibrational average approach and the dynamic one.

The overall good agreement between vibrationally averaged

values and ADMP ones both at room and higher temperature

clearly shows how, for permanganate, a complete dynamic

treatment is not necessary and that less expensive (and fully

quantum) ro-vibrational one-dimensional approaches can be

fruitfully used. Again it should be kept in mind that averaging

along single LAMs is not trivial and is only possible ifmodes are

not coupled and if a suitable vibrational coordinate is chosen.

Acknowledgements

The authors gratefully acknowledge Dr Orlando Crescenzi

(University of Naples) and Dr Philippe Carbonniere (Univer-

sity of Pau) for helpful discussions. We thank Prof. Vincenzo

Barone (University of Naples) for discussions and for

providing a copy of the program Dina.

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