Efficient Calculation of the Rotational g Tensor from Auxiliary Density

Subscriber access provided by UNIV NEW ORLEANS

Article

Efficient Calculation of the Rotational gtensor from Auxiliary Density Functional Theory Bernardo Zúñiga Gutiérrez, Mónica Camacho González, Patricia Simón Bastida, Alfonso Bendaña Castillo, Patrizia Calaminici, and Andreas M. Koster J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp505169k • Publication Date (Web): 26 Jun 2014 Downloaded from http://pubs.acs.org on July 6, 2014

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Efficient Calculation of the Rotational g-Tensor from Auxiliary Density Functional Theory Bernardo Zuniga-Gutierreza,∗ , Monica Camacho-Gonzalezb , Patricia Simon-Bastidab , Alfonso Bendana-Castillob , Patrizia Calaminicia and Andreas M. Köstera,∗

a

Departamento de Qu´ımica, CINVESTAV,

Avenida Instituto Politécnico Nacional 2508 A.P. 14-740 México D.F. 07000, México

b

Departamento de Nanotecnolog´ıa, CIMAV, Av. Miguel de Cervantes 120 Complejo Industrial Chihuahua

Chihuahua, Chihuahua, México C.P. 31109

∗

Electronic e-mails: [email protected], [email protected]

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract

The computation of the rotational g-tensor with the recently developed auxiliary density functional theory (ADFT) gauge including atomic orbital (GIAO) methodology is presented. For the rotational g-tensor, the calculation of the magnetizability tensor represents the most demanding computational task. With the ADFT-GIAO methodology the CPU time for the magnetizability tensor calculation can be dramatically reduced. Therefore, it seems most desirable to employ the ADFT-GIAO methodology also for the computation of the rotational g-tensor. In this work, the quality of rotational g-tensors obtained with the ADFT-GIAO methodology is compared with available experimental data as well as with other theoretical results at the Hartree-Fock and coupled-cluster level of theory. It is found that the agreement between the ADFT-GIAO results and the experiment is good. Furthermore, we also show that the ADFT-GIAO g-tensor calculation is applicable to large systems like carbon nanotube models containing hundreds of atom and thousands of basis functions.

Keywords: ADFT-GIAO, magnetizability, rotational spectroscopy and exchange-correlation potentials

2


Page 2 of 36

Page 3 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


I. Introduction

Since long the rotational g-tensor, also known as molecular g-tensor, has been used for benchmarking first principle molecular magnetic property calculations. The widespread use of this molecular property for benchmarking rest on the availability of very accurate experimental data for a number of small molecules. These data are measured by molecular beams1 or microwave Zeeman spectroscopy.2 The measurements of the related magnetizability property lack sufficient accuracy to be considered seriously for benchmarking a method for magnetic property calculation. The computation of the rotational g-tensor has been performed at various theoretical levels including Hartree-Fock (HF),3–5 MøllerPlesset perturbation theory in second order (MP2),6, 7 third order (MP3)8 and fourth order,9 coupled-cluster single doubles (CCSD), coupled-cluster single-doubles-perturbative-triples [CCSD(T)],10 second-order polarization propagator approximation (SOPPA),11–13 coupledcluster singles-and-doubles polarization-propagator approximation (CCSDPPA),14 secondorder polarization propagator approximation with coupled-cluster singles and doubles amplitudes SOPPA(CCSD)15, 16 multiconfigurational self-consistent-field (MCSCF)17 and, finally, full configuration-interaction (FCI).18 Only recently DFT studies of the rotational g-tensor have emerged.19–21 A reliable prediction of the rotational g-tensor components requires a satisfying treatment of the gauge origin problem. Among several existing schemes, the gauge including atomic orbital (GIAO) scheme has been widely employed since the work of Ditchfield.22 The GIAO scheme ensures gauge-origin independent results and accelerates the basis set convergence.23 In view of computational performance, DFT methods usually offer a reasonable compromise between performance and accuracy for the computation of molecular properties. Further improvement in the performance is achieved by the use of Hermite Gaussian auxiliary densities24 for the variational fitting of the Coulomb potential25–28 and the calculation of the exchange-correlation potential.29 This so-called auxil3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

iary density functional theory (ADFT) approach is well-suited for energy calculations and structures optimizations of small nanosystems with several hundreds of atoms.30–36 Even for these relatively small systems the numerical integration of the exchange-correlation potential in ADFT scales already linearly, yielding a substantial performance improvement with respect to conventional Kohn-Sham DFT methods. In previous works it was shown that the ADFT-GIAO method transfers this performance enhancement to the calculation of magnetic shieldings and magnetizabilities without jeopardizing the accuracy of these molecular property calculations.37, 38 As a result, the ADFT-GIAO approach permits reliable magnetic property calculations on systems with more than 15,000 basis functions and 1,000 atoms in very reasonable times employing moderate computational resources. In this work we extend the ADFT-GIAO methodology to the calculation of the rotational g-tensor.

The article is organized in the following manner. In Sec. II the working equations for the rotational g-tensor calculations within the ADFT-GIAO framework are presented. The computational details for the validation and benchmark calculations are given in Sec. III. The next section, Sec. IV, compares the ADFT-GIAO principal g-tensor components of small molecules with corresponding experimental data as well as other theoretical results obtained at various levels of theory. In Sec. V, benchmark calculations of ADFT-GIAO rotational g-tensors for carbon nanotube models are presented. Finally, concluding remarks are drawn in the last section.

II. Theory

A molecule rotating around its center of mass (com) generates a magnetic moment proportional to its rotational angular momentum.2, 39 The induced magnetic moment due to the rotational motion can interact with an external magnetic field. As a result the degener4


Page 4 of 36

Page 5 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


acy of the rotational states is removed (Zeeman effect). The energy term describing this interaction is commonly expressed as: ∆E = −

µN ~ ~ H · g · J. ~

(1)

~ and J~ denote the external magnetic field and Here µN is the nuclear magneton while H the rotational angular momentum, respectively. For our discussion the most important quantity in Eq. (1) is the so-called rotational g-tensor, g , which is dimensionless. The quantity

µN g ~

is the proportionality tensor between the induced magnetic moment and the

rotational angular momentum. Each element of the g-tensor can be defined as a second derivative of the energy with respect to the Cartesian components of the external magnetic field and the rotational angular momentum: gλη = −

~ ∂ 2E . µN ∂Hλ ∂Jη

(2)

Here ~ is the reduced Planck constant and λ and η denote the Cartesian x, y, and z components. For convenience we switch to atomic units (a.u.) from now on. The nonrelativistic, spin-free Hamiltonian includes the magnetic effects, either due to the external field or due to the induced field by the rotational motion, in the conjugated momentum ˆ . Introducing the rotation of the molecule the expression for this momentum is operator, ~π given as: ˆ = −i∇ ~ +A ~ H (~r ) + A ~ J (~r ) ~π

(3)

~ H and A ~ J denoting the magnetic vector potential contributions arising from the with A external magnetic field and the rotational motion of the molecule, respectively. These vector potentials are defined as:39 ~ × ~r − G ~ ~H = 1 H A 2

(4)

i h J −1 ~ ~ ~ I A = − nuc · J × ~r − Rcom ,

(5)

and

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 36

~ and R ~ com denote the gauge-origin, an arbitrarily chosen respectively. Here the vectors G position, and the center of mass of the molecule, respectively. I nuc is the nuclear tensor of inertia. The observables obtained with this Hamiltonian must be invariant with respect ~ 40 However this gauge invariance only holds when the observables are to the choice of G. obtained from the exact form of the wave function.

In practice, all electronic structure methodologies employ approximated wave functions. This problem can be overcome by using local gauge origin schemes. An overview of these schemes is given in.23 We use the GIAO scheme in the framework of ADFT because it does not require the localization of molecular orbitals as the localized orbital/local origin gauge (LORG)41 or individual gauge for localized orbitals (IGLO)42 schemes. Such a localization procedure can be cumbersome if high computational efficiency is aimed. Furthermore, the GIAO scheme bestow fast basis set convergence for the computation of magnetic properties. In the GIAO scheme, a phase factor, the so-called London factor, is attached to each of the magnetic field independent basis functions, b(~r ).22 However, in order to ensure gauge invariant calculations for the rotational g-tensor, the phase factor must be redefined as follow:4 ~ = b(~r ) e−i(Xb φb (~r, H)

~ H +X~ J )·~ r b

(6)

where 1 ~ ~ H ~ ~ Xb = H × B − G 2

(7)

h i −1 J ~ ~ ~ ~ Xb = − I nuc · J × B − Rcom ,

(8)

and

~ denotes the center of the basis function b(~r ). Including the phase respectively. Here B factor dependent basis functions, Eq. (6), into the ADFT energy expression the following

6


Page 7 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


equation results: E=

X a,b

Pab Hab +

XX a,b

¯ k

1X Pab φa φb ||k¯ − x¯l xk¯ ¯l ||k¯ + Exc [˜ ρ]. 2 ¯¯

(9)

l,k

The gauge-origin independent core Hamiltonian for the rotational g-tensor takes the form: D

ˆ g |b Hab = a |Lab H

E

(10)

with Lab = e 2 H·(A−B )×~r − i(I nuc ·J )·(A−B )×~r

(11)

h i ˆp ~ ˆp ~ −1 ~ ~ ˆ ˆ ~ ~ Hg =H0 + H · ξg (B) + ξg (B) · I nuc · J + h i h i d d −1 −1 ˆ ˆ ~ ~ ~ ~ ~ ~ H · ξ g (B) · I nuc · J − H · ξ (Rcom ) · I nuc · J + . . .

(12)

i

~

~

~

−1

~

~

~

and

ˆ 0 , denotes the unperturbed Hamiltonian: Here the first term, H N 1 ~ 2 X ZC ˆ H0 = − ∇ − ~| 2 r−C C=1 |~

(13)

whereas the second and third term arise from first order perturbations and represent the paramagnetic contributions to the magnetizability and rotational g-tensor, respectively. ˆ ~ Note that the angular momentum operator, ξ~gp (B), appears in both paramagnetic contributions. In the magnetizability contribution it is coupled to the magnetic field whereas in ~ the rotational g-tensor contribution it is coupled to the angular velocity vector, I −1 nuc · J, associated to the rotational state of the molecule. The fourth term arises from second order perturbation and corresponds to the diamagnetic contribution to the rotational g-tensor. The diamagnetic operator has the same form as for the diamagnetic magnetizability contribution but is coupled to the magnetic field and the angular velocity. The fifth term does not arise from the expansion of the conjugated momentum, defined in Eq. (3). Instead it

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 36

has been added to the Hamiltonian in order to describe the interaction between the rotational angular momentum of the electron density of the molecule, rotating as a rigid body, and the external magnetic field. The explicit form of the paramagnetic and diamagnetic operators appearing in Eq.(12) are given by: i ˆ ~ ~ ×∇ ~ ξ~gp (B) = − ~r − B 2 2 d 1 ˆ ~ ~ ~ ~ ξ g (B) = ~r − B E − ~r − B ⊗ ~r − B 8 2 d 1 ˆ ~ ~ ~ ~ ξ (Rcom ) = ~r − Rcom E − ~r − Rcom ⊗ ~r − Rcom . 4

(14) (15) (16)

Here E denotes the unit matrix. All second order terms that are quadratic in the external magnetic field or the angular velocity are omitted in Eq. (12).

The second and third term in the ADFT-GIAO energy expression, Eq. (9), represent the two-electron Coulomb repulsion energy as obtained from the variational fitting of the Coulomb potential.25, 26, 28 For this fitting a linear scaling auxiliary density of the form, ρ˜(~r) =

X

¯ r), xk¯ k(~

(17)

¯ k

¯ r) denotes a primitive Hermite Gaussian auxiliary function24, 43 and is introduced. Here k(~ xk¯ the corresponding density fitting coefficient. These auxiliary functions are perturbation independent in the ADFT-GIAO methodology. They only depend on nuclear and electronic coordinates. Thus, the three-center electronic repulsion integrals within the ADFT-GIAO framework take the following form:

φ∗a φb ||k¯ = Lab ab||k¯ .

(18)

1 . The symbol || is used as shorthand notation for the two-electron Coulomb operator, |~r−~ r′ |

The last term in Eq. (9), corresponds to the exchange-correlation energy which is calculated 8


Page 9 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


from the approximated density, Eq. (17), in the ADFT approach, and from the Kohn-Sham density, that is built from orbitals defined in Eq. (6), in the conventional DFT approach.

Inserting the ADFT-GIAO energy expression into the energy derivative of Eq. (2) and using the explicit form of the core Hamiltonian from Eq. (12) we find the following expression for the (electronic) rotational g-tensor: h i DF T −GIAO d ~ − ξ (Rcom ) I −1 g = −4mp ξ nuc . el

(19)

~ com ) tensor contains the diamagnetic molecular Here mp denotes the proton mass. The ξ d (R integral evaluated with the operator from Eq. (16). The symbol ξ DF T −GIAO denotes the magnetizability tensor as calculated either in the ADFT-GIAO or conventional (Density Fitting) DF-DFT-GIAO methodologies.38

Note that Eq. (19) takes only the electronic contribution into account. Thus, the nuclear contribution must be calculated separately and then added to Eq. (19). Therefore, the final expression for the rotational g-tensor is given by:

g =gg el + g nuc h i ~ com ) I −1 + = − 4mp ξ DF T −GIAO − ξ d (R nuc X 1 ~ ~ 2 ~ ~ ~ ~ ZP P − Rcom E − P − Rcom ⊗ P − Rcom I −1 nuc . 2µN P

(20)

Here P~ and ZP denote the position vector and the nuclear charge of atom P , respectively. For a more detailed derivation of Eq. (20) we refer the interested reader to reference.39

The form of the ADFT-GIAO working equations for the calculation of the rotational gtensor is similar to the one obtained in other theoretical methods.4 In particular, the 9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 36

~ com ), is the same for any one determinant form of the diamagnetic contribution, ξ dia (R approach. Of course the nuclear contribution, g nuc , is also independent from the used electronic structure methodology. The essential difference arises from the magnetizability tensor expression that is significantly simplified in the ADFT-GIAO framework.38 An element of this magentizability tensor is defined as a derivative of the energy with respect to a pair of Cartesian components of the external magnetic field: ∂ 2E . ~ ∂Hη ∂Hλ H→0

ξλη = − lim

(21)

For calculating this energy derivative it is important to note that the response of the electron density vanishes for magnetic field perturbations: ∂ρ(~r) = 0. ~ ∂Hλ H→0

(22)

lim

Thus, it is straightforward to enforce the same condition for the auxiliary density defined in Eq. (17). As a result, the energy derivative in Eq. (20) takes the same form in DF-DFTGIAO and ADFT-GIAO:

ξλη =

X

(λ)

(η)

Pab Hab +

(η)

(λη)

(ηλ)

Pab Hab .

(23)

a,b

a,b

Here Hab and Hab

X

denote the first, and second derivatives of the core molecular integrals, (λ)

respectively. The perturbed density matrix, Pab , is given by McWeeny’s self-consistent perturbation (SCP) theory44 for closed-shell systems as:

(λ) Pab

=2

(λ) (λ) occ X uno X Kju − εj Sju j

u

εj − εu

(caj cbu − cau cbj ) −

1X (λ) Pac Scd Pdb . 2 c,d

(24)

Because of the vanishing (auxiliary) density response with respect to the magnetic field, the perturbed Kohn-Sham matrix is independent of the perturbed density. Thus, no iteration

10


Page 11 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


of Eq. (24) is needed. The ADFT-GIAO and DF-DFT-GIAO approaches differ in the calculation of the perturbed Kohn-Sham matrix elements. In ADFT-GIAO, an element of the perturbed Kohn-Sham matrix in the atomic orbital representation is given as:

(λ)

(λ)

Kab = Hab +

X

¯ (λ) (xk¯ + zk¯ ). hφ∗a φb ||ki

(25)

¯ k

Here the exchange-correlation contributions are taken into account by the so-called exchangecorrelation coefficients, zk¯ , characteristic to ADFT.29 These coefficients are calculated by numerical integration during the self-consistent field (SCF) procedure. Thus, they are not calculated in the magnetic property module.

The corresponding perturbed Kohn-Sham matrix elements in the DF-DFT-GIAO framework have the following form:

(λ) Kab

=

(λ) Hab

+

X

¯ (λ) xk¯ hφ∗a φb ||ki

¯ k

∂ 2 Exc [ρ] + lim . ~ ∂Hλ ∂Pab H→0

(26)

The derivative of the exchange-correlation energy, last term in Eq. (26), yields integrals over GIAOs that need to be solved by numerical integration. The explicit form of these integrals depends on the exchange-correlation approximation used. For the local density approximation (LDA)45 the exchange-correlation energy derivative in Eq. (26) can be written as: LDA ∂ 2 Exc [ρ] = ∂Hλ ∂Pab

Z

∂ǫ(ρ) ∗ (λ) (φa φb ) d~r. ∂ρ

(27)

The generalized gradient approximation (GGA)46–49 depends not only on the electron den~ through γ = ∇ρ ~ · ∇ρ. ~ sity but also on the electron density gradient, ∇ρ, Thus, the corresponding derivative for a GGA exchange-correlation energy functional can be expressed as: 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

GGA ∂ 2 Exc [ρ, γ] = ∂Hλ ∂Pab

Z

∂ǫ(ρ, γ) ∗ (λ) (φa φb ) d~r + 2 ∂ρ

Z

∂ǫ(ρ, γ) ~ ~ ∗ (λ) ∇ρ∇(φa φb ) d~r. ∂γ

Page 12 of 36

(28)

As this discussion shows, the computational demand for the calculation of the ADFTGIAO magnetizability is substantially reduced. Unlike the conventional DFT-GIAO formulation, the ADFT-GIAO magnetizability calculation is free of any numerical integration due to the exchange-correlation contributions. This roots in the different densities used for the calculation of the exchange-correlation energy and potential in the conventional and ADFT Kohn-Sham methods. The exchange correlation fitting coefficients29, 37, 38 in ADFT-GIAO are taken from the previous self-consistent field energy calculation that, of course, includes the corresponding numerical integrations, albeit with perturbation independent basis functions. This is the main reason for the improved computational performance of ADFT-GIAO compared to the conventional DF-DFT-GIAO approaches. Because the ADFT energy is variationally stationary the here described simplification of the magnetizability tensor calculation has no effect on the accuracy of the methodology. Thus, ADFT-GIAO and DF-DFT-GIAO magnetizability tensor are practically indistinguishable. However, the gained computational performance is important if large systems, with hundreds of atoms and thousands of basis functions, are targeted. Also, the dynamical analysis of the rotational g-tensors within the ADFT-GIAO approach studied along a trajectory of a Born-Oppenheimer molecular dynamic calculation becomes feasible in reasonable times. In such studies, the magnetic property must be calculated thousands of times.50

III. Computational details

All calculations were performed with the LCGTO-DFT code deMon2k.51 The exchange12


Page 13 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


correlation potential was either calculated with the Kohn-Sham density or the approximated density. We refer to these methodologies as DF-DFT-GIAO or ADFT-GIAO, respectively. For the comparison of the DF-DFT-GIAO and ADFT-GIAO methods a test set of 70 small molecules for which experimental rotational g-tensor data are reported in the literature was used. All molecular structures of this set were optimized with the generalized gradient approximation (GGA) from Perdew, Burke and Ernzerhof (PBE)49 employing the DZVP-GGA52 basis and GEN-A2* auxiliary function set.52 The rotational g-tensor calculations were performed with the local density approximation (LDA) employing the Dirac exchange53 and VWN correlation functional45 (from now on we refer to this combination as VWN). At the GGA level of theory the exchange-correlation functionals OPTX-PBE48, 49 and OPTX-LYP47, 48 were employed. To investigate the basis set dependency of the rotational g-tensor components the aug-cc-pVXZ basis set family54, 55 was employed. Here X stands for double- (D), triple- (T), quadrupole- (Q) and quintuple-ζ (5). In all rotational g-tensor calculations the GEN-A2 auxiliary function set was used.52

For the comparison of the ADFT-GIAO results with Hartree-Fock and coupled-cluster GIAO results a test set of 25 small molecules for which experimental rotational g-tensor data are available in the literature was employed. The experimental data were taken as reported, i. e. zero point vibrational corrections (ZPVC) are not included. The reason for this omission is the lack of ZPVC data for the here used methodology. For this comparison the exchange-correlation functionals VWN and OPTX-PBE in combination with the aug-cc-pCVQZ basis and GEN-A2* auxiliary function set were used. Optimized CCSD(T)/cc-pVTZ geometries, as provided in the supplementary material of Ref.,56 were used for this set of molecules. To simplify notation we refer to the OPTX-PBE and OPTXLYP functionals as OPBE and OLYP, respectively.

13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 36

As performance benchmarks the rotational g-tensors of single- and double-walled carbon nanotubes as well as for the corresponding peapod structure were calculated. The molecular geometries of these systems were optimized with the semiempirical molecular orbital code MSINDO.57 The energy and property calculations were performed with the deMon2k code using the PBE exchange-correlation functional in combination with the DZVP basis set.58 These calculations were performed using the parallel version of the deMon2k code59–61 on R 8 intel Xeon

TM

cores with 2.4 GHz and 4 GB RAM.

IV. Validation Calculations

In this section the quality of the ADFT-GIAO rotational g-tensors is analyzed. To this end we have calculated mean errors (ME), mean absolute errors (MAE), mean absolute relative errors (MARE), and standard deviations (STD) of the calculated rotational gtensor diagonal elements from the experimental ones. Thus, the errors are defined as: expt calc − gηη error = gηη

(29)

First we investigate how much the ADFT-GIAO rotational g-tensor diagonal elements differ from the ones obtained with the conventional DF-DFT-GIAO methodology. For this study, a test set of 70 small molecules with altogether 151 diagonal elements measured with high precision is employed. The individual ADFT-GIAO and DF-DFT-GIAO rotational g-tensor diagonal elements are compared with their experimental counterparts in Tables SI-I to SI-IV of the supporting information. Table I summarizes the errors, standard deviations, correlation coefficients and linear regression slopes of the calculated ADFT-GIAO and DF-DFT-GIAO rotational g-tensor diagonal elements with respect to experiment for the test set. The MEs, MAEs and MAREs obtained with the ADFT-GIAO framework are very similar to the ones from the conventional DF-DFT-GIAO. The same holds for the 14


Page 15 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


correlation coefficients, R2 , and the slopes of the linear regression. Thus, we can conclude from Table I that the quality of the calculated rotational g-tensors with the ADFT-GIAO and DF-DFT-GIAO methodologies is practically identical, independent of the used basis set and exchange-correlation functional.

As the data in Table I show, the quality of the calculated rotational g-tensor diagonal elements usually improves with basis set size. Note that the used Dunning basis sets improve systematically the correlation to experiment with increasing basis set size without being especially optimized for magnetic property calculations. The comparison of the LDA and GGA quality of the rotational g-tensor diagonal elements in Table I shows a systematic and significant improvement at GGA level of theory. The overall best agreement in terms of MEs, MAEs and MAREs with experiment is achieved with the OPBE exchangecorrelation functional in combination with the largest aug-cc-pV5Z basis set. Obviously, this very large basis set will considerably impact the computational performance of the ADFT-GIAO methodology if applied to large systems. Because the MEs and MAEs in Table I increase only moderately with decreasing basis set sizes, high quality rotational g-tensors can already be obtained with rather moderate basis sets of double or triple zeta valance plus polarization quality. This permits reliable rotational g-tensor calculations of very large systems with the ADFT-GIAO methodology.

Table II compares ADFT-GIAO VWN and OPBE rotational g-tensors with their counterparts from Hartree-Fock, HF, and coupled-cluster methods, CCSD and CCSD(T), as well as with experimental results. For this comparison a test set of 25 molecules with 47 rotational g-tensor diagonal elements, all measured with high precision, is employed. The HartreeFock and coupled-cluster data are taken from reference.56 All rotational g-tensors were calculated with the aug-cc-pCVQZ basis set. Table III summarizes the errors, standard 15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

deviations, correlation coefficients and linear regression slopes of the calculated rotational g-tensors from the various theoretical methods listed in Table II with respect to experiment. A marked difference between the ADFT and ab-initio results is seen in the mean error (ME). Whereas for ab-initio methods the mean error is positive, it is negative for the ADFT approaches. This indicates that the ADFT-GIAO approaches usually underestimate the rotational g-tensor diagonal components while the ab initio methods overestimate them. The comparison of the MARE data in Table III shows that VWN and HF methods yield a similar quality for the rotational g-tensor elements. Similarly, the OPBE and CCSD methods show very similar performance with respect to the experimental results. These two methods improve significantly the quality of the rotational g-tensor components with respect to the VWN and HF methods and come close to the CCSD(T) reference data. In Figure 1 VWN and OPBE rotational g-tensor diagonal elements are correlated to their experimental counterparts. As this figure shows the correlation is for both approaches good for most diagonal elements. Usually, the OPBE results are in better agreement to experiment.

V. Benchmark Calculations

As already mention, the ADFT-GIAO methodology is free of numerical integration which bestow a high performance to the method. To show this in more detail we depict in Figure 2 the serial timings for the calculation of the rotational g-tensor of tetracene (C18 H12 ) with the conventional DF-DFT-GIAO (light gray bars) and the ADFT-GIAO (dark gray bars) methodology. The basis sets used, with the number of basis functions enclosed by parenthesis, are aug-cc-pVDZ (558), aug-cc-pVTZ (1290) and aug-cc-pVQZ (2550). The optimized PBE/DZVP-GGA/GEN-A2 geometry was used. Figure 2 shows that even for the small basis set with only 558 basis functions, a notable improvement in CPU timing is achieved 16


Page 16 of 36

Page 17 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


with the ADFT-GIAO methodology. It is 10 times faster! The performance difference increases with increasing basis set size. For the largest basis set with 2550 basis functions, the ADFT-GIAO calculation is 34 times faster than its conventional counterpart. As a result, the 29 CPU hours enduring DF-DFT-GIAO rotational g-tensor calculation is performed in less than 1 hour with the ADFT-GIAO methodology. Despite this enormous reduction in CPU time the resulting rotational g-tensor elements are very similar, as expected from our previous validation calculations. Table IV lists the calculated diagonal components of the rotational g-tensor in the principal axis orientation for the tetracene molecule.

Finally, we employed our ADFT-GIAO implementation to compute the rotational g-tensors to three models of carbon nanotubes: single-walled, peapod and double-walled carbon nanotubes. The number of atoms and basis functions, enclosed in parenthesis, for each model are 640 (9200), 820 (11900) and 1024 (14720), respectively. Figure 3 shows the parallel timings for the self-consistent field (dark gray bar) and the magnetizablity tensor (light gray bar) calculations. The timings for the rotational g-tensor (black bar), excluding the timing for the magnetizability tensor, is also presented. As Figure 3 shows the SCF procedure is by far the computationally most expensive task representing around 40% of the total time for all three systems. The computation of the magnetizability tensor takes around 8% of the total time. This tensor is the main ingredient to compute the rotational g-tensor. The ~ com ) and computational time needed for the computation of the diamagnetic tensor ξ d (R the nuclear contribution, g nuc , are negligible as Figure 3 shows.

The calculated diagonals components of the rotational g-tensor in the principal axes orientation of the carbon nanotubes are given in Table V. The components of the three models follow the relation gxx < gyy = gzz reflecting the high symmetry of the models. As table V shows the rotational g-tensor diagonal elements of the single walled carbon nanotube are 17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

larger in magnitude than for the double-walled and the peapod nanotubes. As the rotational g-tensors depend on the inverse of the tensor of inertia, molecules trapped inside a carbon nanotube decrease the ability of the system to rotate and also the ability to induced a magnetic moment. To the best of our knowledge this represents the first report of the rotational g-tensors in the literature for these kind of systems.

VI. Conclusions

In this paper the calculation of rotational g-tensors within the ADFT-GIAO methodology is presented. We demonstrate that the quality of ADFT-GIAO rotational g-tensors is practically indistinguishable from the ones of conventional DF-DFT-GIAO calculations. Despite these similar qualities, the computational timings are very different. Benchmark calculations show that the ADFT-GIAO methodology is orders of magnitudes more efficient than the conventional DF-DFT-GIAO method for rotational g-tensor calculations.

Our validation calculations show that the best agreement with experimental data is achieved with GGA functionals and large basis sets. In particular, the quality of the ADFT-GIAO OPBE/aug-cc-pCVQZ/GEN-A2* rotational g-tensor compares favourable with the one from CCSD calculations. As a result, the ADFT-GIAO methods permits reliable rotational g-tensor calculations of small nanosystems with several hundreds of atoms. As show case applications the rotational g-tensors of single- and double-walled as well as peapod carbon nanotubes were calculated. In these applications the system size ranges up to around 1000 atoms with more than 14000 basis functions.

Acknowledgments

18


Page 18 of 36

Page 19 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


This work was supported by the CONACYT projects CB-179409 and CB-130726. The authors gratefully acknowledge computational time on the Xiuhcoatl hybrid clusters of CINVESTAV, Mexico, and HPC resources of the WestGrid, Canada.

Supporting Information

The individual ADFT-GIAO and DF-DFT-GIAO rotational g-tensor diagonal elements are available in Tables SI-I to SI-IV of the supporting information. This information is available free of charge via the Internet at http://pubs.acs.org.

19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

References [1] Ramsey, N. Molecular Beams; Oxford University: New York, 1956. [2] Flygare, W. H. Magnetic Interactions in Molecules and an Analysis of Molecular Electronic Charge Distribution from Magnetic Parameters. Chem. Rev. 1974, 74, 653–687. [3] Kelly, H. M.; Fowler, P. W. Quadrupole Moments, Magnetisabilities and Rotational g Factors for First- and Second-row Hydrides. Chem. Phys. Lett. 1993, 206, 568–573. [4] Gauss, J.; Ruud, K.; Helgaker, T. Perturbation–Dependent Atomic Orbitals for the Calculation of Spin–Rotation Constants and Rotational g Tensors. J. Chem. Phys. 1996, 105, 2804–2812. [5] Ruud, K.; Helgaker, T. The Magnetizability, Rotational g Tensor, and Quadrupole Moment of PF3 Revisited,. Chem. Phys. Lett. 1997, 264, 17–23. [6] Cybulski, S. M.; Bishop, D. M. Calculations of Magnetic Properties. IV. Electron– Correlated Magnetizabilities and Rotational g Factors for Nine Small Molecules. J. Chem. Phys. 1994, 100, 2019–2026. [7] Cybulski, S. M.; Bishop, D. M. Calculations of Magnetic Properties. II. Electron– Correlated Nuclear Shielding Constants for Nine Small Molecules. J. Chem. Phys. 1993, 98, 8057–8064. [8] Cybulski, S. M.; Bishop, D. M. Calculation of Magnetic Properties. VI. Electron Correlated Nuclear Shielding Constants and Magnetizabilities for Thirteen Small Molecules. J. Chem. Phys. 1997, 106, 4082–4090. [9] Sauer, S. P. A.; Jensen, H. J. A.; Ogilvie, J. F. Quantum–Chemical Calculations of Radial Functions for Rotational and Vibrational g Factors, Electric Dipolar Moment 20


Page 20 of 36

Page 21 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and Adiabatic Corrections to the Potential Energy for Analysis of Spectra of HeH+ . Adv. Quantum Chem. 2005, 48, 319–334. [10] Gauss, J.; Ruud, K.; Kállay, M. Gauge–Origin Independent Calculation of Magnetizabilities and Rotational g Tensors at the Coupled–Cluster Level. J. Chem. Phys. 2007, 127, 074101. ˇ [11] Sauer, S. P. A.; Spirko, V.; Oddershede, J. The Magnetizability and g–Factor Surfaces of Ammonia. Chem. Phys. 1991, 153, 189–200. [12] Sauer, S. P. A. A Relation Between the Rotational g-Factor and the Electric Dipole Moment of a Diatomic Molecule. Chem. Phys. Lett. 1998, 297, 475–483. [13] Ogilvie, J. F.; Oddershede, J.; Sauer, S. P. A. The Rotational g Factor of Diatomic Molecules in State 1 Σ+ or 0+ . Adv. Chem. Phys. 2007, 111, 475–536. [14] Sauer, S. P. A.; Ogilvie, J. F. Experimental and Theoretical Estimates of the Rotational g Factor of AlH in the Electronic Ground State X1.SIGMA.+. J. Phys. Chem. 1994, 98, 8617–8621. [15] Sauer, S. P. A. Theoretical Estimates of the Rotational g-Factor, Magnetizability and Electric Dipole Moment of GaH. Chem. Phys. Lett. 1996, 260, 271–279. [16] Sauer, S. P. A. The Rotational g Tensor of HF, H2 O, NH3 , and CH4 : A Comparison of Correlated Ab Initio Methods. Adv. Quantum Chem. 2005, 48, 469–490. [17] Ruud, K.; Helgaker, T.; Jørgensen, P. The Effect of Correlation on Molecular Magnetizabilities and Rotational g Tensors. J. Chem. Phys. 1997, 107, 10599–10606. [18] Ruud, K.; ˚ Astrand, P.-O.; Helgaker, T.; Mikkelsen, K. V. Full CI Calculations of the Magnetizability and Rotational g Factor of the Hydrogen Molecule. J. Mol. Struct. (Theochem) 1996, 388, 231–235. 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

[19] Wilson, P. J.; Amos, R. D.; Handy, N. C. Accurate Predictions for Molecular Magnetisabilities and Rotational g Tensors Using a Simple and Efficient DFT Approach. J. Mol. Struct. (Theochem) 2000, 506, 335–342. [20] Wilson, D. J. D.; Mohn, C. E.; Helgaker, T. The Rotational g Tensor as a Benchmark for Density-Functional Theory Calculations of Molecular Magnetic Properties. J. Chem. Theory Comput. 2005, 1, 877–888. [21] Lutnæs, O. B.; Teale, A. M.; Helgaker, T.; Tozer, D. J. Rotational g Tensors Calculated Using Hybrid Exchange-Correlation Functionals with the Optimized Effective Potential Approach. J. Chem. Theory Comput. 2006, 2, 827–834. [22] Ditchfield, R. Molecular Orbital Theory of Magnetic Shielding and Magnetic Susceptibility. J. Chem. Phys. 1972, 56, 5688–5691. [23] Gauss, J. In Modern Methods and Algorithms of Quantum Chemistry, Vol. 1, NIC Series; Grotendorst, J., Ed.; John von Neumann Institute for Computing: J¨ ulich, Germany, 2000. [24] Köster, A. M. Hermite Gaussian Auxiliary Functions for the Variational Fitting of the Coulomb Potential in Density Functional Methods. J. Chem. Phys. 2003, 118, 9943–9951. [25] Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. On First-Row Diatomic Molecules and Local Density Models. J. Chem. Phys. 1979, 71, 4993–4999. [26] Mintmire, J. W.; Dunlap, B. I. Fitting the Coulomb Potential Variationally in LinearCombination-of-Atomic-Orbitals Density-Functional Calculations. Phys. Rev. A 1982, 25, 88.

22


Page 22 of 36

Page 23 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[27] Dunlap, B. I.; Rösch, N.; Trickey, S. B. Variational Fitting Methods for Electronic Structure Calculations. Mol. Phys. 2010, 108, 3167–3180. [28] Mintmire, J. W.; Sabin, J. R.; Trickey, S. B. Local-Density-Functional Methods in TwoDimensionally Periodic Systems. Hydrogen and Beryllium Monolayers. Phys. Rev. B 1982, 26, 1743–1753. [29] Köster, A. M.; Reveles, J. U.; del Campo, J. M. Calculation of Exchange-Correlation Potentials with Auxiliary Function Densities. J. Chem. Phys. 2004, 121, 3417–3424. [30] Salahub, D. R.; Calaminici, P.; Gamboa, G. U.; Köster, A. M.; Vásquez, J. M. In Metallic systems: A Quantum Chemist’s Perspective; Allison, T.; Coskuner, O.; González, C. A., Eds.; (CRC Press): Boca Raton. Florida, 2011. [31] Calaminici, P.; Dominguez-Soria, V. D.; Flores-Moreno, R.; Gamboa, G. U.; Geudtner, G.; Goursot, A.; Salahub, D. R.; Köster, A. M. In Handbook of Computational Chemistry; Leszczynski, J., Ed.; Springer-Verlag: Berlin Heidelberg, Germany, 2011. [32] Geudtner, G.; Calaminici, P.; Carmona-Esp´ındola, J.; Dominguez-Soria, V. D.; FloresMoreno, R.; Gamboa, G. U.; Goursot, A.; Köster, A. M.; Reveles, J. U.; Mineva, T.; et al. Wiley Interdisciplinary Reviews: Computational Molecular Science 2012, 2, 548–555. [33] Calaminici, P.; Geudtner, G.; Köster, A. M. First-Principle Calculations of Large Fullerenes. J. Chem. Theory Comput. 2009, 5, 29–32. [34] Calaminici, P.; Carmona-Espindola, J.; Geudtner, G.; Köster, A. M. Static and Dynamic Polarizability of C540 Fullerene. Int. J. Quantum Chem. 2012, 112, 3252–3255.

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

[35] Dominguez-Soria, V. D.; Calaminici, P.; Goursot, A. Theoretical Study of the Structure and Properties of Na-MOR and H-MOR Zeolite Models. J. Chem. Phys. 2007, 127, 154710. [36] Dominguez-Soria, V. D.; Calaminici, P.; Goursot, A. Theoretical Study of Host-Guest Interactions in the Large and Small Cavities of MOR Zeolite Models. J. Phys. Chem. C 2011, 115, 6508–6512. [37] Zuniga-Gutierrez, B.; Geudtner, G.; Köster, A. M. NMR Shielding Tensors from Auxiliary Density Functional Theory. J. Chem. Phys. 2011, 134, 124108. [38] Zuniga-Gutierrez, B.; Geudtner, G.; Köster, A. M. Magnetizability Tensors from Auxiliary Density Functional Theory. J. Chem. Phys. 2012, 137, 094113. [39] Sauer, S. P. A. Molecular Electromagnetism: A Computational Chemistry Approach; Oxford Univ. Press: New York, 2011. [40] Landau, L. D.; Lifschitz, E. M. Lehrbuch der theoretischen Physik. 2: Band II, Klassische Feldtheorie; Akademie-Verlag: Berlin, 1992. [41] Hansen, A. E.; Bouman, T. D. Localized Orbital/Local Origin Method for Calculation and Analysis of NMR Shieldings. Applications to

13

C Shielding Tensors. J. Chem.

Phys. 1985, 82, 5035–5047. [42] Kutzelnigg, W. Theory of Magnetic Susceptibilities and NMR Chemical Shifts in Terms of Localized Quantities. Isr. J. Chem. 1980, 19, 193–200. [43] Alvarez-Ibarra, A.; Köster, A. M. Double Asymptotic Expansion of Three-center Electronic Repulsion Integrals. J. Chem. Phys. 2013, 139, 024102.

24


Page 24 of 36

Page 25 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[44] Dodds, J. L.; McWeeny, R.; Sadlej, A. J. Self-Consistent Perturbation Theory Generalization for Perturbation-Dependent Non-orthogonal Basis Set. Mol. Phys. 1977, 34, 1779–1791. [45] Vosko, S. H.; Wilk, L.; Nusair, M. Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: a Critical Analysis. Can. J. Phys. 1980, 58, 1200–1211. [46] Becke, A. D. Density-Functional Exchange-energy Approximation with Correct Asymptotic Behavior. Phys. Rev. A 1988, 38, 3098–3100. [47] Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785–789. [48] Handy, N. C.; Cohen, A. J. Left-Right Correlation Energy. Mol. Phys. 2001, 99, 403– 412. [49] Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. [50] Goursot, A.; Mineva, T.; Vásquez-Pérez, J. M.; Calaminici, P.; Köster, A. M.; Salahub, D. R. Contribution of High-Energy Conformations to NMR Chemical Shifts, a DFTBOMD Study. Phys. Chem. Chem. Phys. 2013, 15, 860–867. [51] Koster, A. M.; Geudtner, G.; Calaminici, P.; Casida, M. E.; Dominguez, V. D.; FloresMoreno, R.; Gamboa, G. U.; Goursot, A.; Heine, T.; Ipatov, A.; et al. deMon2k, Version 3, The deMon developers, Cinvestav, México, D.F. 2011. [52] Calaminici, P.; Janetzko, F.; Köster, A. M.; Mejia-Olvera, R.; Zuniga-Gutierrez, B. Density Functional Theory Optimized Basis Sets for Gradient Corrected Functionals: 3d Transition Metal Systems. J. Chem. Phys. 2007, 126, 044108. 25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

[53] Dirac, P. A. M. Note on Exchange Phenomena in the Thomas-Fermi Atom. Proc. Cambridge Philos. Soc. 1930, 26, 376–385. [54] Dunning, T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. [55] Woon, D. E.; Dunning, T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. III. The Atoms Aluminum Through Argon. J. Chem. Phys. 1993, 98, 1358–1371. [56] Lutnæs, O. B.; Teale, A. M.; Helgaker, T.; Tozer, D. J.; Ruud, K.; Gauss, J. Benchmarking Density-Functional-Theory Calculations of Rotational g Tensors and Magnetizabilities Using Accurate Coupled-Cluster Calculations. J. Chem. Phys. 2009, 131, 144104. [57] Bredow, T.; Geudtner, G.; Jug, K. MSINDO Parameterization for Third-Row Transition Metals. J. Comput. Chem. 2001, 22, 861–887. [58] Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Optimization of GaussianType Basis Sets for Local Spin Density Functional Calculations. Part I. Boron Through Neon, Optimization Technique and Validation. Can. J. Phys. 1992, 70, 560–571. [59] Geudtner, G.; Janetzko, F.; Köster, A. M.; Vela, A.; Calaminici, P. Parallelization of the deMon2k Code. J. Comput. Chem. 2006, 27, 483–490. [60] Calaminici, P.; Dominguez-Soria, V. D.; Geudtner, G.; Hernández-Mar´ın, E.; Köster, A. M. Parallelization of Three-Center Electron Repulsion Integrals. Theor. Chem. Acc. 2006, 115, 221–226.

26


Page 26 of 36

Page 27 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


[61] Calaminici, P.; Dominguez-Soria, V. D.; Geudtner, G.; Hernández-Mar´ın, E.; Köster, A. M. Parallelization of the Variational Fitting of the Coulomb Potential. Superficies y Vac´ıo 2005, 18, 1–5.

27



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 36

Table I: Errors (ME, MAE, MARE), standard deviations (STD), correlation coefficients (R2 ) and linear regression slopes for the calculated ADFT-GIAO and DF-DFT-GIAO rotational g-tensors with respect to experiment and empirical equilibrium data. See text for details. ME ADFT DF-DFT

MAE ADFT DF-DFT

D T Q 5

-0.0095 -0.0094 -0.0095 -0.0086

-0.0099 -0.0103 -0.0106 -0.0109

0.0149 0.0145 0.0142 0.0138

0.0155 0.0146 0.0145 0.0143

D T Q 5

-0.0075 -0.0091 -0.0088 -0.0076

-0.0070 -0.0070 -0.0073 -0.0074

0.0104 0.0105 0.0101 0.0085

0.0100 0.0098 0.0098 0.0098

D T Q 5

-0.0030 -0.0041 -0.0035 -0.0017

-0.0035 -0.0033 -0.0035 -0.0036

0.0099 0.0089 0.0085 0.0080

0.0092 0.0081 0.0079 0.0078

MARE ADFT DF-DFT VWN 11.4772 9.9110 9.7101 9.6136 10.7024 9.8236 10.3471 9.8123 OLYP 10.9553 12.4430 9.9811 10.7964 13.8015 12.3294 14.3844 11.8506 OPBE 8.3777 8.0220 8.3277 8.7118 7.6175 8.1577 7.5645 8.1264

STD ADFT DF-DFT

R2 ADFT DF-DFT

ADFT

Slope DF-DFT

0.0573 0.0559 0.0551 0.0530

0.5383 0.5073 0.5263 0.5476

0.9989 0.9991 0.9991 0.9992

0.9989 0.9993 0.9993 0.9994

0.9072 0.9085 0.9098 0.9129

0.9070 0.9126 0.9109 0.9091

0.0233 0.0156 0.0155 0.0136

0.1507 0.1640 0.1644 0.1545

0.9981 0.9990 0.9990 0.9992

0.9985 0.9989 0.9989 0.9990

1.0108 1.0088 1.0087 1.0024

1.0115 1.0156 1.0164 1.0158

0.0302 0.0242 0.0242 0.0241

0.1711 0.1604 0.1530 0.1540

0.9976 0.9985 0.9986 0.9987

0.9979 0.9987 0.9989 0.9990

1.0281 1.0244 1.0260 1.0270

1.0080 1.0124 1.0122 1.0131

28


Page 29 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table II: Comparison of ADFT-GIAO (VWN and OPBE), Hartree-Fock-GIAO and coupled-cluster-GIAO rotational g-tensors with experimental data. The aug-cc-pCVQZ basis set was used in all calculations. HF CO 15 N 2 H2 O

HC15 N HOF

NH3 CH2 O

CH4 AlF CH3 F C3 H 4

FCCH FC15 N H2 S

HCP HFCO

H 2 C2 O

LiF LiH N2 O OCS OF2

H 4 C2 O

SO2

VWN 0.7530 -0.2874 -0.2788 0.6621 0.6764 0.7354 -0.1082 -0.1271 -0.0724 0.6870 0.5269 0.5926 -3.4369 -0.2406 -0.1095 0.3584 -0.0801 -0.0686 0.2904 -0.1606 -0.0961 0.0563 -0.0100 -0.0525 0.2320 0.2661 0.4249 -0.0543 -0.4856 -0.0825 -0.0386 -0.3133 -0.0367 -0.0246 0.0555 -0.6584 -0.0825 -0.0301 -0.2547 -0.0758 -0.0655 -0.1038 0.0185 0.0351 -0.6542 -0.1235 -0.0893

OPBE 0.7296 -0.2901 -0.2786 0.6319 0.6502 0.7086 -0.1077 -0.1225 -0.0690 0.6605 0.4956 0.5621 -3.1126 -0.2359 -0.0961 0.3254 -0.0836 -0.0628 0.2558 -0.1480 -0.0968 0.0545 -0.0081 -0.0519 0.1937 0.2378 0.4033 -0.0515 -0.4700 -0.0803 -0.0374 -0.3459 -0.0349 -0.0250 0.0573 -0.6044 -0.0801 -0.0287 -0.2420 -0.0728 -0.0628 -0.0930 0.0201 0.0350 -0.6716 -0.1222 -0.0866

HFa 0.7627 -0.2816 -0.2699 0.6641 0.6834 0.7323 -0.0773 -0.0951 -0.0440 0.7011 0.5072 0.5780 -2.7019 -0.2227 -0.0664 0.3033 -0.0842 -0.0550 0.2680 -0.1484 -0.0939 0.0627 -0.0032 -0.0479 0.1253 0.1894 0.3804 -0.0313 -0.4137 -0.0769 -0.0355 -0.4041 -0.0336 -0.0253 0.0733 -0.6963 -0.0786 -0.0285 -0.1429 -0.0610 -0.0471 -0.0807 0.0360 0.0382 -0.6766 -0.1201 -0.0882

CCSDa 0.7535 -0.2669 -0.2576 0.6550 0.6725 0.7281 -0.0846 -0.1062 -0.0561 0.6863 0.5084 0.5755 -2.8013 -0.2175 -0.0856 0.3209 -0.0801 -0.0586 0.2731 -0.1402 -0.0800 0.0606 -0.0055 -0.0483 0.1778 0.2234 0.3945 -0.0359 -0.4228 -0.0767 -0.0362 -0.4322 -0.0341 -0.0244 0.0691 -0.6647 -0.0775 -0.0277 -0.1846 -0.0642 -0.0525 -0.0876 0.0287 0.0371 -0.5927 -0.1160 -0.0868

a

Values taken from reference.56

b

the aug-cc-pCV[TQ]Z basis set was used.

c

As given in.56

CCSD(T)a 0.7527 -0.2678 -0.2591 0.6547 0.6707 0.7284 -0.0882 -0.1110 -0.0601 0.6840 0.5093 0.5757 -2.8659 -0.2197 -0.0924 0.3231 -0.0794 -0.0606 0.2711 -0.1431 -0.0803 0.0592 -0.0063 -0.0487 0.1861 0.2300 0.3974 -0.0385 -0.4271 -0.0771 -0.0368 -0.4297 -0.0347 -0.0245 0.0677 -0.6638 -0.0780 -0.0280 -0.1977 -0.0672 -0.0551 -0.0910 0.0259 0.0366 -0.5985 -0.1165 -0.0869

CCSD(T)a,b Expt.c 0.7542 0.7416 -0.2681 -0.2689 -0.2591 -0.2593 0.6563 0.6450 0.6717 0.6570 0.7303 0.7180 -0.0882 -0.0904 -0.1107 -0.1190 -0.0596 -0.0610 0.6853 0.6420 0.5107 0.5024 0.5770 0.5654 -2.8641 -2.9017 -0.2194 -0.2243 -0.0910 -0.0994 0.3236 0.3133 -0.0794 -0.0805 -0.0601 -0.0620 0.2724 0.2650 -0.1424 -0.1492 -0.0802 -0.0897 0.0595 0.0536 -0.0062 -0.0077 -0.0487 -0.0504 0.1883 0.1950 0.2321 0.2090 0.3978 0.3550 -0.0382 -0.0430 -0.4268 -0.4227 -0.0770 -0.0771 -0.0367 -0.0371 -0.4300 -0.4182 -0.0345 -0.0356 -0.0243 -0.0238 0.0678 0.0737 -0.6649 -0.6584 -0.0778 -0.0789 -0.0279 -0.0288 -0.1972 -0.2130 -0.0671 -0.0680 -0.0550 -0.0580 -0.0905 -0.0946 0.0267 0.0189 0.0369 0.0318 -0.5988 -0.6043 -0.1165 -0.1163 -0.0873 -0.0887

29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 36

Table III: Errors (ME, MAE, MARE), standard deviations (STD), correlation coefficients (R2 ) and linear regression slopes for the calculated rotational g-tensors of Table II with respect to experiment. See text for details. Comparinson with experimental data ME MAE MARE

VWN

OPBE

HFa

−0.0083

−0.0066

0.0089

0.0086

0.0060

0.0299

0.0171

0.0194

0.0104

0.0076

6.72

4.80

10.12

6.26

CCSDa

11.79

CCSD(T)a

STD

0.0836

0.0374

0.0370

0.0172

0.0108

R2

0.9974

0.9988

0.9980

0.9996

0.9998

Slope

0.8780

0.9526

1.0300

1.0148

1.0002

a

From reference.56

Table IV: Calculated components of the rotational g-tensor for tetracene employing the PBE/aug-cc-pVXZ/GEN-A2 methodology (X=D,T,Q). The shadowed rows correspond to the ADFT-GIAO results while the white rows correspond to the DF-DFT-GIAO ones.

aug-cc-pVDZ

aug-cc-pVTZ

aug-cc-pVQZ

gxx

gyy

gzz

-0.064595

-0.009607

0.012279

-0.065850

-0.009692

0.013214

-0.065285

-0.009785

0.014766

-0.066901

-0.009930

0.015260

-0.065040

-0.009732

0.015582

-0.067120

-0.009986

0.015755

30


Page 31 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table V: Calculated components of the rotational g-tensor for the studied carbon nanotubes. The calculations were performed with the PBE/DZVP/A2 methodology. gxx

gyy

gzz

Single walled

-0.008869

0.001618

0.001618

Peapod

-0.008857

0.001169

0.001169

Double walled

-0.006526

0.000962

0.000962

31



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure captions

Figure 1: Comparison of the calculated ADFT-GIAO VWN and OPBE rotational g-tensor diagonal elements with experimental data. The data are taken from Table II.

Figure 2: Comparison of CPU timings (min) between the ADFT-GIAO and DF-DFTGIAO methodologies for the calculation of the rotational g-tensor of tetracene with the aug-cc-pVDZ, aug-cc-pVTZ and aug-cc-pVQZ basis.

Figure 3: CPU timings for the rotational g-tensor calculations of single-walled, peapod and double-walled carbon nanotubes. All calculations were performed with the PBE/DZVP/A2 R methodology. Timings refer to 8 Intel XeonTM cores with 2.4 GHz and 4GB RAM.

32


Page 32 of 36

Page 33 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1

Figure 1: B. Zuniga-Gutierrez et al. 33



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2



Page 34 of 36

Page 35 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

36


Page 36 of 36

Efficient Calculation of the Rotational g Tensor from Auxiliary Density

Recommend Documents