Charge Equilibration Based on Atomic Ionization in MetalâOrganic

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Charge Equilibration Based on Atomic Ionization in Metal-Organic Frameworks Brad Arthur Wells, Caspar de Bruin-Dickason, and Alan L. Chaffee J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp510415h • Publication Date (Web): 02 Dec 2014 Downloaded from http://pubs.acs.org on December 6, 2014

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The Journal of Physical Chemistry

Charge Equilibration Based on Atomic Ionization in Metal-Organic Frameworks Brad A. Wells*, Caspar Debruin-Dickason, Alan L. Chaffee* CRC for Greenhouse Gas Technologies, School of Chemistry, Monash University, Melbourne, Australia. Charge Equilibration, Grand canonical Monte Carlo, Metal-organic framework, Carbon Dioxide

Abstract In this paper two new charge equilibration methods for estimating atomic partial charges are outlined. These methods are based on expanding the Taylor series used to estimate the ionization energy of each atom around either the formal or atomic charge, allowing for accurate charge estimation in both covalent and ionic materials.

A new treatment of

hydrogen atoms based on molecular hydrogen is also introduced. To demonstrate the general applicability of these new methods they have been applied to the simulation of CO2 adsorption in Metal-Organic Frameworks. Comparisons between other charge equilibration methods and Density Functional Theory (DFT) calculations show that of the rapid charge assigning methods, the algorithm based on atomic ionization best replicates the DFT electrostatic potential and provides the most accurate estimation of CO2 adsorption.

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1. Introduction Porous coordination networks (PCNs) or Metal-organic frameworks (MOFs) are a relatively new class of materials that are currently being investigated for a wide range of applications1. One area in particular where their use is advocated is in the area of gas storage and separation. Their typically large pore volume and selective adsorption make them suited to areas such as carbon capture2 and hydrogen storage3. Molecular simulation can play an important role in understanding the capabilities of MOF materials4. Simulations can both lead to insights into gas adsorption that are not generally available experimentally,5 as well as help to screen large databases of MOFs to help find MOFs for particular applications6. Simulations of gas adsorption in MOF materials commonly employ Grand Canonical Monte Carlo (GCMC) techniques7. In these simulations the energy of interaction between gas molecules and the framework is usually represented by a combination of dispersion, repulsion and electrostatic potentials such that the energy E is given as:

,

4

(1)

The dispersion and repulsion potentials and respectively are commonly taken from empirical forcefields designed to replicate particular data. Some common forcefields used for gases and frameworks are the Universal8, Dreiding9, TraPPE10 and OPLS-AA11 forcefields. These potentials are usually transferable across atom types and for screening


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calculations reasonable results can be obtained using simple atom classifications. The atomic partial charges are more difficult to assign from simple atom types. Although there have been some attempts12, atomic charges are usually not transferable in the same way as van der Waals (VDW) potentials, as they often depend on the polarization of charge across the entire molecule13. The relatively strong electrostatic forces, especially in materials with highly polar exposed metal cations, make the assignment of partial charges a key step in these types of GCMC simulations.4b To address the need to find realistic atomic partial charges several different types of methods have been employed. One of the most common methods is to generalize the results of high level electronic Density Functional Theory (DFT) theory to the assignment of charges. In MOF materials calculations on both isolated sections of the framework and also the entire periodic unit cell have been used. These calculations can be generalized in a number of ways. Firstly the calculated wavefunction can be used to infer charges by using schemes such as Mulliken charge assignment14. While usually well defined, these methods have a tendency to underestimate the polarization of charge in the framework.15 Secondly estimation of the electrostatic interactions between the gas and framework can be based on the calculated electrostatic potential. Least squares fitting procedures such as the ChelpG16 for molecules and REPEAT17 for periodic cells have been developed for this purpose. While these methods can produce charges that do a reasonable job of replicating the electrostatic potential these algorithms can suffer from numerical instability and can produce charges that are chemically unrealistic. This is especially true of materials that have buried atoms that do not directly contribute to the electrostatic potential around the surface of the pores. To overcome these problems the fitting procedure may use a set of model charges which bias the solution towards chemically sensible charges. Another possible use for the


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calculated electrostatic potential is to simply use the potential without fitting charges directly in the GCMC calculation.18 A third class of methods for assigning charges are the methods that partition the molecular electron density into contributions from each atom. The Bader Quantum Theory of Atoms In Molecules (QTAIM) method19 based on dividing the electron density by surfaces of zero flux is one of the most well studied methods. Recently some new methods such as the DDEC method15, 20 and the iterative Hirshfeld21 method have been applied to assigning partial atomic charges in periodic systems. These methods partition the electron density by using reference atomic electron densities for different ionization states.

In these methods charges are

assigned iteratively as the calculated charges alter the reference atomic electron density. One final important class of methods is the electronegativity equalization method22. In these methods partial charges are assigned through a simple consideration of atomic ionization potentials. Assuming that the electronegativity of each atom in the molecule is equivalent and that the molecule has a set overall charge leads to a series of linear equations that can be solved for the atomic charges. The original idea for this method was expressed by Sanderson23. Initial formulations of this method suffered from unrealistic charges due to treating the electrostatic interactions between atoms at close range as point charges. This lead to the use of effective electronegativity and atomic hardness values fitted to reproduce experimental or calculated results24. Subsequent improvements were made by Rappe and Goddard in the QEq method25, who suggested the use of electrostatic repulsion integrals and charge dependent parameters for hydrogen atoms. The QEq method was later generalized to include periodic cells26. Recently the QEq method was modified for use in calculating partial charges in MOF materials27. The main modification in these methods is the expansion of the Taylor series approximation of the atomic ionization energies around the ionization states of the metal cations. Kadantsev and coworkers have also recently parameterized the QEq


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method for MOF materials, in a way that is reminiscent of older electronegativity equalization methods.28 While these recent extended QEq methods do perform better than the original QEq method for MOFs, some improvements to the general methodology is still desirable. In this paper we describe two new methods for calculating the partial atomic charges. These methods are based around different approaches to expanding the Taylor series and also different treatments of the hydrogen atom. In this paper we define these new methods for calculating atomic partial charges. We then test these methods by applying them to estimating the low pressure CO2 adsorption in a range of MOF materials. The results of these calculations are compared both to experiment and calculations of CO2 adsorption using a range of other partial charge assignment methods.

2. Theory General method The general aim of charge equilibration methods is to balance the ionization energies of different atoms with the electrostatic interactions that result from the polarization of charge in the molecule. One of the central approximations assumptions in this method is that the energy of each atom, (), as a function of charge can be approximated by a Taylor series expansion, such that:

( ) ( )

1 ⋯ ( ) 2

(2)

The derivatives of energy can be related to atomic ionization potentials (IP) and electron affinities (EA). So for instance, when expanding the Taylor series around the energy of the neutral atom, the derivatives of energy can be given as:

"# $ ! 2

(3)


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"# % !

(4)

Here $ is the atomic electronegativity and % represents the atomic hardness. Combining the equations for the energy of each atom truncated at the quadratic term gives a total energy of the molecule &(')

&(')

(

(

)

1 ) * (+ ) ( ) 2

(5)

Here * (+ ) is a function that describes the electrostatic interactions between pairs of

atoms. Taking the derivative of energy with respect to charge produces the atomic chemical potential , , such that

&(' ) ) ( * (+ ) ,

(6)

Equating each of the atomic chemical potentials produces N-1 independent linear equations. The final necessary equation is found by constraining the overall charge of the molecule to a particular value -./0 such that:

-./0

(7)

The charges are most efficiently found through solving the matrix equation:

12 3

(8)

Here the elements of the square matrix 1 are given as

4 5 (

) * 7+ 8 * 7+ 8 for < ) 5 (

≤> 1

6

4 1

(9)

(10)


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The elements of the column matrix 3 are likewise given as

?

for < ≤ > 1 @ 6

6

? -./0

(11) (12)

Solving for the column matrix 2 produces the charges for each of the N atoms as elements of the matrix.

Expansion of the Taylor series In formulating a charge equilibration method one of the choices that needs to be made is the set of charges about which the Taylor series approximation of the atomic ionization energy is expanded.

In the original formulations of the electronegativity equalization

methods as well as the QEq method the Taylor series is expanded around the derivatives of the neutral atom. This is the most obvious choice for covalently bonded materials. It also has the advantage of easily connecting atomic properties with the required derivatives of energy. However, as pointed out by Wilmer and coworkers, this approximation is likely to be a poor one in ionic bonded materials such as MOFs.27b In these materials the charge on some atoms is far from neutral, leading to inaccuracies in the approximation of the ionization energy.

To counteract this they suggested, in their Extended QEq (E-QEq) method,

expanding the ionization energy around the ionic charge for metal cations and around the neutral state for other atoms.27a While leading to more accurate ionization energies for metal cations this approach has the downside of artificially introducing a background positive charge into the unit cell. In the E-QEq method the effect of this background charge is minimized through the use of an empirically determined dielectric constant. In order to avoid this background charge it is necessary to expand the atomic ionization energies in a systematic way that includes both positive and negative charges. One way to do this is to expand the Taylor series for each atom around the formal charge of each atom. This


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leads to the same expansions for metal cations but also includes negative expansions for the organic atoms in the MOF linkers that bear negative charge. Fractional formal charges, such as those that can be found on the nitrogen atoms in a typical Zeolitic Imidazole Framework (ZIF) can be handled through a linear interpolation of derivatives such that:

(1 A) ( A) B BC

Here A is the value of

(13)

rounded down to the nearest integer value. This interpolation of

derivatives to accommodate a continuous change in the reference charge is similar to the partitioning of the reference density in the iterative Hirschfeld method.29 We label this new method for determining the reference charges the ‘Formal Charge Equilibration method’ (FC-QEq). While the FC-QEq method is consistent in its assignment of the expansion charges, application of this method in large scale screening is still somewhat problematic. While atomic formal charge can be automatically calculated based on simple atom connectivity rules, ionization states of some metals can be more difficult determine. This is a problem that is shared with the E-QEq method as well. For large scale screening it may be useful to include the calculation of the expansion charges as part of the calculation rather than as input. To achieve this one possible approach is to assume that in an ideal solution the expansion charges and the atomic charges are equal. The atomic partial charges and the Taylor series expansion charges can then be found iteratively in a self-consistent fashion. This ensures that the expansion for each atom is appropriate without needing information about ionization states or formal charges. It should also present no problems for structures that have illdefined bonding and for which the usual empirical rules used to find formal charge do not work. We label this second method for finding the reference charges the ‘Ionizing Charge

Equilibration method’ (I-QEq) as here the molecule is considered as a collection of ions rather than atoms.


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Atomic Ionisation Energies To expand the Taylor series approximation of the ionization energy several energies for each atom around the typical ionization states are required. In the E-QEq method these ionization energies are taken from experiment. While this produces high quality data for atoms taking on positive charges it does not provide comparable data for atoms taking on negative charges.

In contrast to ionization energies it is difficult to measure electron

affinities past the first ionization state. For both the FC-QEq and I-QEq methods, which require a description of electron affinities of negatively charged atoms, the necessary experimental data is not available. To provide data for our new QEq methods we have chosen to use electron affinities and ionization potentials from high level quantum mechanics calculations.

The energies of

several ionization states for each atom have been calculated for each atom up to krypton and also palladium. These calculations were performed using the quantum chemistry package Gaussion0930. For each calculation the CCSD(T) method combined with the augmented ccpVQZ basis set was used. Atomic spin states were chosen so as to minimize the energy of each ionization state.

These calculations produce relative energies that are generally

comparable with those from experiment. A full listing of the calculated atom energies is given in the supporting information. From this ionization data first and second derivatives of energy can be calculated using simple three point derivative formulas. Attempts at using higher order polynomials or cubicsplines to estimate derivatives were found to be too numerically unstable. It is also worth noting that the calculated ionization energies allow for estimating higher order derivatives for an extended Taylor series approximation. However while possible, the accuracy of these higher order derivatives is likely to be insufficient to improve the overall accuracy of the Taylor series expansion. As an illustration of this a QEq scheme with third and fourth order terms was recently parameterized by Oda and Takahasi on the basis of fitting to calculated


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RESP charges.31

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Comparison of their fitted higher order derivatives and derivatives

calculated from the calculated ionization energies shows significant differences.

Electrostatic interactions

In order to define the overall energy of the molecule &(') it is necessary to define the

function * 7+ 8 that describes the energy of interaction between two charges. In the original charge equilibration methods atoms were considered as point charges and this function was just given as a simple Coulomb potential such that:

* 7+ 8

1

D+ D

(14)

While computationally simple, this function leads to an over-estimation of the interaction energy of two closely spaced atoms. A second level of approximation, first suggested in the QEq method is to assume that the charge on each atom is spread out over an atomic orbital

E (+ ) such as a Slater orbital.25 The electrostatic interaction is then calculated through the Coulomb integral

* 7+ 8 F

E (+ )E (+ ) D+ + D

+ +

E (+ ) >|+ |@H I HJ |+|

(15) (16)

Due to the numerical complexity of evaluating these integrals over Slater functions several alternate approaches have been suggested, such as using empirical approximations to the integral and also using simpler functions to describe the atomic charge, such as Gaussian functions.32 In all our charge equilibration methods we have approximated the Slater integral using the DasGupta-Huzinaga equation

* 7+ 8

1

H 1 D+ D 2 K* I LD+M D * I LD+M D N

(17)


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Here O is the Klondike parameter33 which, following Oda and Hirono, we take to be 0.4. *

and * are given from the Slater integral in equations (15)-(16) centered on the ith and jth atoms respectively. In the Slater function in equation (16), the parameter P effectively defines the size of each atom.

In the original QEq paper the P parameter for each atom was fitted using the

relationship

〈 〉

(2W 1) S +TE (+)U + V 2P STE (+)U +

(18)

Here the average radius 〈 〉 was estimated from crystal structure data, which was then used

to estimate P . V was considered a scaling parameter that accounted for the difference in the definitions of average atom size. In the QEq method V was fit to 0.5 through calculation of metal-halide dipoles. In both the FC-QEq and I-QEq methods atoms may be treated as either neutral or charged,

and thus the appropriate P parameter should change with the ionization state of the atom. To estimate appropriate atomic radii for both atoms and ions the electron density for each case was calculated.

This was again achieved using Gaussian09.30

Electron densities were

calculated using the UHF method with a double zeta basis set. This density was then tabulated on a three dimensional grid and the average radius was calculated by numerical integration.

Radii for partial ionization states were calculated through simple linear

interpolation. Consideration was also given to the scaling factor V. In the FC-QEq method the original QEq value of 0.5 was retained. In the I-QEq method a larger value of 0.8 was found empirically to give better results. The need for different scaling factors may be accounted for in that in the FC-QEq method the atomic radius is evaluated at the formal charge of the atom. In the I-QEq method the atomic radius changes is allowed to vary with


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the calculated charge on the atom, and thus it is appropriate for the atom radius to more closely match that of the isolated ion. This is achieved in increasing the scaling parameter. In applying the charge equilibration method to periodic materials such as MOFs some account also needs to be taken of the lattice potential.

This is most simply done by

calculating the entire electrostatic lattice interaction X 7+ 8 and then replacing closely spaced terms with the more accurate integral terms. This leads to a general equation

* 7+ 8 X 7+ 8 F

E (+ )E (+ ) D+ D

+ +

1

D+ D

(19)

One accurate method of calculating the lattice energy suggested in a several methods is through Ewald summation34 such that:

X 7+ 8 \

erfc([D+ \D) D+ \D

5 [ 4 _ _ I H|^| /ab cos(^ ∙ + ) Ω

(20)

^

Here \ represents the lattice vectors, ^ the reciprocal lattice vectors, and [ is the convergence parameter.

One less computationally demanding alternative to Ewald

summation is the shifted-force potential,35

X 7+ 8 e \

erfc([D+ \D) D+ \D

erfc([gh ) 2αeHb jk ) f (D+ \D gh )mn gh √gh _ _

(21)

In our calculations we used Ewald summation for the calculation of the lattice energy.

Hydrogen As noted by Rappe and Goddard, hydrogen is a problematic atom to treat accurately in charge equilibration methods. The underlying cause of this is the difference in the shape of the 1s orbital in atomic hydrogen and the sigma bonding orbital in a covalently bound hydrogen atom. In the case of the 1s atomic orbital the shape of the orbital is more easily perturbed to accommodate the addition of an electron, resulting in a high electron affinity. In covalently bound hydrogen atoms there is not the same orbital flexibility and therefore the


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electron affinity is much lower. This difference in electron affinity leads to poor results if the atomic parameters are used without modification in the charge equilibration procedure. In order to overcome these differences in atomic and covalent hydrogen in their QEq method Rappe and Goddard25 treated both the atomic hardness and the orbital size of hydrogen atoms as charge dependent. Their equations of the atomic hardness of hydrogen %o

and the orbital size of hydrogen Po were given as

%o () p1

q % Po o

Po () Po

(22) (23)

The ionization potential and electron affinity of hydrogen were also both modified empirically to produce meaningful charges. These were given the values of 11.47eV and 2.41eV respectively. To overcome the problems associated with empirically fitted parameters a new and simplified method for hydrogen was developed. This approach to approximating charge transfer to hydrogen atoms is based on the ionization energies of the hydrogen molecule, H2, rather than on the hydrogen atom. In the hydrogen molecule the bonding and anti-bonding orbitals are similar to those in a typically covalently bonded hydrogen atom.

Since a

hydrogen molecule was considered the relevant reference state, here negative ion energies are generalized as half of the energy of the H22n- ionization states. The atomic energy of the H+ state is retained. In this way the hydrogen molecule is considered as a pseudo-atom for negative hydrogen ion. Calculations on the hydrogen molecule produced an electron affinity of 2.62eV. This compares well with the fitted values in the original QEq method as well as the fitted electron affinity of 2eV used in the E-QEq method27a, and gives some theoretical justification to what is in the E-QEq method an ad-hoc parameter. Here, in the FC-QEq and I-QEq methods, we use the calculated electron affinities for the hydrogen molecule. Both these methods are therefore non-iterative with respect to finding the charges given a


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particular expansion of the Taylor series. The I-QEq method still requires iteration to find the appropriate expansion of the Taylor series. This approach also has the benefit that it avoids the use of fitted ad-hoc parameters, and therefore remains generally applicable to a wide range of materials. In particular the FC-QEq and I-QEq methods do not require a scaled dielectric constant. The requirement of the EQEq method for a scaled dielectric constant can prove problematic in large scale screening studies. For instance in one recent study an increased dielectric constant of 2 was used, presumably to produce realistic charges for the large 130,000 structure screening set.36 This large dielectric constant, while increasing the numerical stability of the method, also lead to an under-estimation of the electrostatic interactions within the framework, and therefore an underestimation of CO2 adsorption as well.

3. Testing Charge Assignment Methods MOF Structures To create a test set of applicable MOF structures we have selected 24 different MOFs encompassing a range of geometries, structural motifs and metals. MOFs that have available CO2 adsorption data were selected. The common names, metal center and selected CO2 adsorption temperature of each MOF is listed in TABLE 1. For comparison to simulation, the CO2 adsorption of each MOF at the listed temperature was estimated from the data in the literature. Interpolations of the 0.1 bar and 1 bar adsorption values were performed by fitting isotherm equations to the data. Where available isotherms focused on the 0-1 bar pressure range were used. It should be noted that the experimental estimation of CO2 adsorption itself is not a straightforward task. While some duplicated measurements of gas adsorption in the same MOF show similar results, this is not true in all cases, such as in the case of Cu3(BTC)2.37 A number of factors, including activation procedure can influence the gas adsorption in some MOFs.38 In most cases however the most significant variation between


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isotherms is in the gas adsorption at higher pressures. As the isotherm approaches the maximum loading the adsorption is governed mainly by the available pore volume, and may be influenced by the pore topology and long-range order of the material. In this study, where we have focused on low pressure adsorption, the experimental isotherms form a generally reliable estimate of gas adsorption. The estimated experimental CO2 adsorption is shown graphically in Figure 1.

Figure 1. Estimated experimental CO2 loading in test MOFs From this data the range of CO2 adsorption in MOFs can be seen. At 0.1 bar the adsorption of the selected MOF materials varies over two orders of magnitude. Even at 1 bar the adsorption of CO2 still varies over an order of magnitude. Interestingly the adsorption of CO2 at the lower 0.1 bar pressure point does not obviously correlate with adsorption at the higher pressure of 1 bar. This is because at low pressure adsorption is dominated by the effect of CO2 interactions with the preferred binding sites. At higher pressure the interactions with the framework become less important and the available pore volume also becomes important. This change in adsorption mechanism is most clearly seen in the difference between the adsorption of the small pore Pd(2-pymo)2 and the much larger pore Zn-MOF-74 and Cu3(BTC)2. Overall this variance in the adsorption data illustrates the challenges associated with accurate prediction of CO2 adsorption in MOF materials.


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To create a model of each MOF for simulation the starting unit cells and atomic positions in each of these MOFs were taken from the literature. The atomic positions were then optimized using DFT calculations carried out using the package Siesta39.

In these

calculations the PBE functional40 with D2 VDW corrections was employed41. An optimized single zeta numerical basis set and a plane wave cutoff of 200 Ry was also used.

Charge Assignment Methods To determine how the FC-QEq and I-QEq methods compare to other available methods for charge assignment several different charge assignment methods were applied to the MOFs in the test set. Firstly DFT calculations using Siesta were used to assign charges to each atom. Here single point calculations, similar to those used in geometry optimization were employed using an optimized double zeta polarized numerical basis set. Both Mulliken charges and REPEAT17 charges were fitted using the calculated density matrix and electrostatic potential. This electrostatic potential was also used directly as a reference potential. Secondly the Connectivity-Based Atom Contribution (CBAC) method was also applied to each MOF.12a, 12b

This method assigns charges based on reference calculations of molecules with similar

atomic types. Thirdly a range of charge equilibration methods, the QEq, E-QEq, FC-QEq and I-QEq were used to assign partial charges to the atoms. These calculations were done with our own code, using the parameters from the Materials Studio program42 for the QEq method and in the literature for the E-QEq method. Finally electrostatic interactions were ignored in one series of calculations. In applying the CBAC method to the MOFs in the test set, not all MOFs contained atoms types that are included in the charges of the CBAC method. Therefore to extend the CBAC methodology to these frameworks additional charges were calculated using a procedure similar to that used in the original papers.12a,

12b

Single-point molecular DFT calculations

were carried out on molecular fragments cut out from the larger MOF crystal. Charges were


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calculated using a combination of the B3LYP density functional, the cc-pVTZ and LanL2TZ basis sets43 and ChelpG methods. Charges from atoms of the same type in different MOFs were averaged across different structures. The new charges were then added into the CBAC charge set. Charges from the CBAC method were linearly adjusted so that the overall charge neutrality of the unit cell was maintained.

Adsorption Calculations To test the applicability of each method to simulations of carbon capture, the adsorption of CO2 in each framework was modeled. Here GCMC simulations were used to estimate the CO2 loading of each material at pressures of 0.1 and 1 bar. The simulations were carried out using a code developed within our group. In the Monte Carlo routines addition, deletion, rotation, translation and regrowth steps were all considered. A cavity bias method44 was used to accelerate convergence of the simulation. Simulation cells were constructed of copies of the unit cell so that simulation cells were at least 25Å in each direction. Simulations used 2×106 cycles to load the material and 2×106 cycles to estimate the gas loading. Temperatures in each case were selected so as to match the available experimental data. In the case of Pd(2-pymo)2 gas insertion into the small cages within the structure was prohibited as these cages are kinetically inaccessible to adsorbing CO2. CO2 was modeled using three-point potentials, with each C-O bond being 1.16Å in length. Interactions between gas molecules were estimated using potentials from the TraPPE forcefield.10 Following this forcefield charges of -0.35 and +0.7 were assigned to the oxygen and carbon atoms respectively.

Both VDW and electrostatic interactions were calculated

using an accelerated Ewald method with an error limit of 1×10-3 kcal mol-1. Dispersion and repulsion interactions between the gas and the MOF material were estimated with potentials from the Dreiding9 and Universal8 forcefields on the framework and potentials from the TraPPE forcefield10 for CO2. The Universal forcefield was used for atoms without a relevant


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type defined in the Dreiding forcefield. Lorentz-Berthelot combination rules were used to assign parameters for each pairwise interaction. Both electrostatic and VDW interactions were pre-calculated on three dimensional grids and interpolated for each energy evaluation. Ewald summation with error limits of 1×10-5 kcal mol-1 was used to calculate the grids, each of which had grid point spacing of approximately 0.15Å in each direction. In the case of simulations using the DFT based electrostatic potential, this potential was directly imported into the simulation from Siesta.

4. Results and Discussion Comparisons of Electrostatic Potentials One measure of quality of different charge assignment schemes is their ability to replicate the calculated electrostatic potential of a material.

Indeed replicating the electrostatic

potential is the basis for a number of least-squares charge fitting procedures. Therefore to test the various charge assignment methods used here the average difference between the DFT and charge assigned electrostatic potentials was calculated.

Here the measure of

distance used is the Relative Root Mean Square (RRMS) difference, which for electrostatic potentials can be given as

RRMS u

∑+(Ewx. (+) Ey (+) E )

∑+7Ewx. (+)8

(24)

In this expression + represents points on the three dimensional grid of electrostatic

potential, Ewx. (+) represents the DFT based electrostatic potential, Ey (+) is the charge fitted atomic based electrostatic potential and E is the zero of the DFT based electrostatic

potential. In these measurements E is chosen so as to minimize the distance between the two electrostatic potentials. For the comparison only points that were within 2 VDW radii of

any atom and no closer than 0.5 VDW radii to all atoms were used, where the VDW radii were taken from the Universal forcefield. This method provides a more direct comparison


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between the DFT electrostatic potential and the potential from assigned point charges as it avoids the fitting of charges, essentially combining charge fitting and comparison into one calculation. The results of these calculations are shown graphically in Figure 2. Here the standard deviation of each RRMS average is shown in the error bar, along with the average and maximum RRMS for each method.

Figure 2. RRMS difference between DFT and charge assigned electrostatic potentials From this figure it can be seen that of the difference charge assignment methods the REPEAT method produces the electrostatic potential closest to that of DFT, which is unsurprising given that these charges are specifically fitted to replicate the potential. Of the charge equilibration based methods the I-QEq method has the lowest average, standard deviation and maximum RRMS, indicating that it has the best overall performance in replicating the electrostatic potential of the rapid charge assignment schemes. Interestingly the I-QEq method approaches the Mulliken method in approximating the electrostatic potential, at only a fraction of the computational time. The FC-QEq produces a lower average RRMS difference but a larger maximum difference than the E-QEq method. The CBAC method produces a similar average RRMS to the E-QEq, although with higher standard deviations and maximum RRMS. This suggests that this method is, in general, less reliable than the charge equilibration based methods.


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Finally it can be seen that QEq method fails quite badly in many cases in providing a reasonable estimate of the electrostatic potential. This is due in part to the poor performance of the QEq method on copper based MOFs. Using the QEq method the charge assigned to copper was in many cases greater than +2e-. As previously noted there is some discrepancy in the QEq parameters for copper.28 Unfortunately the atomic parameters required to apply the QEq method, and also the related Universal forcefield, to transition metal compounds remains unpublished. The parameters used in our calculations were taken from the Materials Studio program.42 Other values for these parameters have also been suggested.45

Comparisons of adsorption simulations Another method for evaluating the applicability of charge equilibration methods is to compare the results of molecular simulations using these methods.

Here GCMC gas

adsorption simulations have been used to evaluate the selected charge assignment methods. In these simulations the energy contains both VDW and electrostatic contributions. It is therefore instructive to compare simulations using different charge assignment methods to simulations using the reference electrostatic potential and to experiment. Comparisons to simulations using the reference electrostatic potential allow the isolation of the effect of different descriptions of the charge within the framework. Comparison to experiment allows the overall accuracy of each method to be evaluated. The distinction is important as VDW interactions comprise a significant proportion of the adsorption energy in most MOF materials. Comparisons to experimental adsorption values require an accurate treatment of both VDW and electrostatic interactions, and thus it may be possible, in some cases, to obtain accurate adsorption values through the cancellation of errors in these two components. The adsorption of CO2 in each framework was simulated at 0.1 and 1 bar employing each of the representations of the electrostatic potential.

A comparison between simulated


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adsorptions using the DFT based electrostatic potential and various atom based potentials is shown in Figure 3.

Figure 3. Comparison between simulated CO2 adsorptions Here from this scatter plot the range of data for the different methods can be seen. As expected, ignoring the electrostatic interactions tends towards underestimation of the adsorption. Conversely the CBAC method tends towards overestimating the adsorption. Comparing the charge equilibration methods the original QEq method is clearly the most unreliable, the FC-QEq method tends towards over-estimating the adsorption and the E-QEq method shows a significant scattering of adsorption results. The I-QEq method follows relatively closely the DFT calculated adsorption, as does the REPEAT and Mulliken methods. This is in line with our previous calculations of the RRMS difference between the DFT and charge assigned electrostatic potentials. To further quantify the performance of each method the Mean Relative Absolute Difference (MRAD) difference between both the DFT based and experimental adsorption was calculated for each method. Here the MRAD is defined as

zg4

{ | 2 ∑

D D > { |

(25)


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Page 22 of 50

where { is the adsorption of CO2 for material < estimated by method }. The central reason for using this measure of difference is that it is not dominated by a large difference by a single value in the set. The results of this analysis are shown in Figure 4.

Figure 4. Mean relative absolute difference between simulated CO2 adsorptions From this data it can be seen that the original QEq method is quite unsuited to calculations on MOF materials. Here the QEq method produced in most cases less accurate adsorption than simply ignoring the framework electrostatic interactions entirely. The CBAC performs relatively poorly in comparison to the DFT based simulations. One significant source of error in the CBAC method is the inclusion of new atom types into the set of charges. In general charges for different atom types within a set need to be well balanced to achieve reasonable electrostatics. This is achieved by balancing the polarization of charge between a number of contributions from different atoms within the original training set so that the relative charges calculated for adjacent atoms match well.

Unbalanced charges in this

context can lead to areas of artificially large charge or artificial dipoles. The charges added in our set in some cases upsets the sum of the balance in the original set, which leads to poor charge assignment in these cases. This highlights the difficulty of extending the CBAC method for large scale computational screening.


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Focusing on the comparisons to the adsorption simulations using the DFT derived electrostatic potential, it can be seen that the REPEAT method gives quite similar results. This highlights that, given the ability to use the DFT electrostatic potential directly, REPEAT charges are in general redundant. Of the charge equilibration methods the I-QEq method shows significantly better results than the E-QEq and FC-QEq methods. This is significant as it shows that the increased fidelity to the DFT potential shown in Figure 2 translates into a similar estimated adsorption. Moving on to comparisons between the calculated adsorption and experiment, it can be seen that of all the methods the I-QEq and FC-QEq methods produce the lowest mean relative absolute differences in adsorption. This is initially a surprising result as it would be expected that a DFT based method would be the most accurate. However, looking across the methods it can be seen that no single method produces a highly accurate estimate of adsorption in all cases, especially at low pressure. This is the same observation of Wilmer et. al. in their work on the E-QEq method.27a Indeed some other screening studies on MOFs have shown cases where standard GCMC simulations have produced substantively different CO2 adsorptions than those observed experimentally, especially for the M-MOF-74 materials.46 One substantial source of error in these calculations that has not been taken into account is the impact of the VDW parameters used to estimate the dispersion and repulsion potentials. In these simulations, as in a number of other studies we have used parameters from the standard Dreiding and Universal forcefields. While these potentials are simple to apply to a range of materials, they generally do not give accurate adsorption energies in a number of important cases, including the case of open metal sites.47 While there recently has been a large interest in calculating accurate potentials for use in gas adsorption simulations,48 most of these methods are too computationally demanding for a large scale screening study. This


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result highlights the need for methods for dispersion and repulsion potentials that work reliably with MOFs. The individual contributions of the electrostatic and VDW potentials on the adsorption can be further analyzed by comparing the components of the calculated adsorption energies of CO2. In Figure 5 the percentage of electrostatic energy in the total gas-framework energy at 0.1 bar is shown. This data is sorted in descending order for the DFT field calculation.

Figure 5. Percentage of electrostatic energy the total gas-framework energy at 0.1 bar It can be seen that in nearly all simulations the electrostatic energy is the minor component of adsorption energy.

In more than half the tested materials the electrostatic potential

contributions are less than 10% of the total gas-framework potential calculated using the DFT field and REPEAT methods. Electrostatic interactions do play a significant in the adsorption of some materials. However for a number of materials the electrostatic interactions are quite low, and therefore adsorption is driven mostly by dispersion interactions between the gas and the framework. In these cases the quality of the VDW potentials determines the quality of the adsorption calculation. Another interesting conclusion that can be drawn from this data is the ability of each method to estimate the electrostatic contribution to the adsorption energy.

This electrostatic

adsorption energy for probe molecules such as CO2 can be seen as a measure of the intrinsic


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charge polarization in the MOF material. While the magnitudes of the electrostatic energies do differ, the I-QEq method is able to produce the same distinction between polar and nonpolar MOFs as found by the DFT field and REPEAT methods. The E-QEq and FC-QEq method do not produce the same separation of materials into polar and non-polar. This is another indicator of the overall reliability of the I-QEq method. In screening MOF test sets for CO2 adsorption it is often more desirable to estimate the relative ordering of the gas adsorption rather than the absolute gas adsorption. Given that the purpose of screening is to isolate the materials with the most promise for further study, the correct ordering of adsorption allows this determination. Therefore to further characterize the performance of each method the Spearman and Pearson correlation coefficients were calculated for each method.49 The Spearman coefficient is a measure of how well each method predicts the ordering of gas adsorption in each framework. The Pearson correlation coefficient is a measure of the covariance of two data sets, and is an estimate of how well each method predicts the absolute adsorption. The results of these calculations are shown in Figure 6.


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Figure 6. Spearman and Pearson correlation coefficients for each method The 0.1 bar experimental Spearman correlation coefficients are calculated as 0.70, 0.79 and 0.80 for the E-QEq, FC-QEq and I-QEq methods respectively. This compares well with the similar value of 0.73 for the E-QEq method given by Wilmer and co-workers for their test set of MOFs.27a The 0.1 bar DFT Spearman and Pearson correlation coefficients of 0.98 and 0.78 respectively for the I-QEq method does not compare as favorably with the value of 0.98 given for both factors by Kadantsev at coworkers for the their MOF Electrostatic Potential Optimized (MEPO) QEq method.28 The difference in the Pearson correlation factors is likely due to the higher pressure of 0.15 bar used in their comparisons and the smaller range of metals used in their test set. Indeed they identify that the accuracy of their method is reduced when applied to open-metal vanadium MOFs outside of their original test set.


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In general these correlation coefficients show the applicability of the I-QEq method to adsorption screening calculations, especially at lower pressures.

Of the rapid charge

assignment methods it produces the largest correlation coefficients at 0.1 bar compared against the DFT field simulations. In comparison to experiment, while the CBAC method produced a slightly larger Spearman correlation factor than the I-QEq method, the low Pearson correlation factor for the CBAC method highlights the relatively poor estimation of absolute adsorption with this method. The slightly better performance of the I-QEq method at estimating CO2 adsorption compared to the DFT based methods is best explained by the observation that as the I-QEq method improves the estimation of electrostatic interactions in the simulations such that the major source of error becomes the approximate VDW potential. While some of the accuracy of the I-QEq method in estimating experimental CO2 adsorption could potentially be produced by a fortuitous cancellation of errors between the electrostatic and VDW potentials, the close similarity between the DFT and I-QEq electrostatic potentials shows that this is unlikely a large effect.

5. Conclusions Two new charge equilibration methods for estimating the electrostatic interactions within MOF materials, the FC-QEq and I-QEq were outlined. These methods are based both on the appropriate expansion of the Taylor series used to estimate the atomic energy around the ionization state of the atom, and also new treatments of the ionization of molecular hydrogen which avoids fitted parameters. These methods have been applied to a range of different MOFs, and compared to other methods for estimating electrostatic interactions.

These

comparisons showed that the I-QEq method provides a better estimation of the electrostatic potential than the other tested charge equilibration methods.

This replication of the

electrostatic potential also leads to more accurate estimations of CO2 adsorption than other rapid charge assignment methods. Comparisons of simulated and experimental isotherms


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reveal the VDW potential to be a major source of error in the simulation. This highlights the need for more advanced estimation of the VDW potential to compliment the improved electrostatic interactions provided by the I-QEq method.

Author Information Corresponding Author *E-mail: [email protected] (B.A.W); [email protected] (A.L.C.)

Present Address School of Chemistry, Monash University, Victoria 3800, Australia.

Acknowledgements The authors acknowledge the support provided by the Australian Government through its CRC Program for this CO2CRC research project.

Supporting Information Available The calculated ionisation energies, atomic ion sizes, CO2 adsorption isotherms and calculated correlation coefficients are provided in the supporting information. The source code for a program that illustrates the calculation of FC-QEq and I-QEq charges is also provided. This material is available free of charge via the Internet at http://pubs.acs.org.

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21 (9), 3070-3076. 57. Demessence, A.; D'Alessandro, D. M.; Foo, M. L.; Long, J. R., J. Am. Chem. Soc. 2009, 131 (25), 8784-8786. 58. Banerjee, R.; Phan, A.; Wang, B.; Knobler, C.; Furukawa, H.; O'Keeffe, M.; Yaghi, O. M., Science 2008, 319 (5865), 939-943. 59. Liang, Z.; Marshall, M.; Chaffee, A. L., Microporous Mesoporous Mater. 2010, 132 (3), 305-310. 60. Llewellyn, P. L.; Bourrelly, S.; Vagner, C.; Heymans, N.; Leclerc, H.; Ghoufi, A.; Bazin, P.; Vimont, A.; Daturi, M.; Devic, T.; Serre, C.; Weireld, G. D.; Maurin, G., J. Phys.

Chem. C. 2012, 117 (2), 962-970. 61. Furukawa, H.; Yaghi, O. M., J. Am. Chem. Soc. 2009, 131 (25), 8875-8883.


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62. Saha, D.; Bao, Z.; Jia, F.; Deng, S., Environ. Sci. Technol. 2010, 44 (5), 1820-6. 63. Yazaydin, A. O.; Snurr, R. Q.; Park, T.-H.; Koh, K.; Liu, J.; LeVan, M. D.; Benin, A. I.; Jakubczak, P.; Lanuza, M.; Galloway, D. B.; Low, J. J.; Willis, R. R., J. Am. Chem. Soc. 2009, 131 (51), 18198-18199.


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TABLE 1: MOFs contained in test set MOF ID

Common

Metal

Temperature /K

Reference

Name 1

Mg-MOF-74

Mg

296

50

2

Co-MOF-74

Co

296

50

3

Ni-MOF-74

Ni

296

50

4

Zn2(NDC)2(DPNI) Zn2

296

51

5

Zn-MOF-74

Zn

296

50

6

Pd(2-pymo)2

Pd

293

52

7

Cu3(BTC)2

Cu2

295

53

8

ZIF-78

Zn

298

54

9

PCN-11

Cu2

300

55

10

PCN-6

Cu2

298

56

11

CuCl-HBTTri

Cu2Cl

298

57

12

ZIF-69

Zn

298

58

13

Zn2(BDC)2(TED)

Zn2

298

59

14

ZIF-68

Zn

298

58

15

MIL-47

V

303

60

16

Ni2(BDC)2(TED)

Ni2

298

59

17

COF-103

B

273

61

18

COF-102

B

273

61

19

MOF-5

Zn4O

295

62

20

ZIF-8

Zn

298

63

21

IRMOF-3

Zn4O

298

63

22

UMCM-1

Zn4O

298

63

23

MOF-177

Zn4O

295

37a

24

PCN-16

Cu2

300

55


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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 ACS Paragon Plus Environment

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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

Graphic Abstract 298x157mm (96 x 96 DPI)


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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. Estimated experimental CO2 loading in test MOFs 57x39mm (300 x 300 DPI)



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. RRMS difference between DFT and charge assigned electrostatic potentials 57x39mm (300 x 300 DPI)


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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. Comparison between simulated CO2 adsorptions 57x39mm (300 x 300 DPI)



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 4. Mean relative absolute difference between simulated CO2 adsorptions 57x39mm (300 x 300 DPI)


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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 5. Percentage of electrostatic energy the total gas-framework energy at 0.1 bar 57x39mm (300 x 300 DPI)



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 6. Spearman and Pearson correlation coefficients for each method 117x167mm (300 x 300 DPI)


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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. Estimated experimental CO2 loading in test MOFs 256x178mm (96 x 96 DPI)



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. RRMS difference between DFT and charge assigned electrostatic potentials 256x178mm (96 x 96 DPI)


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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. Comparison between simulated CO2 adsorptions 256x178mm (96 x 96 DPI)



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 4. Mean relative absolute difference between simulated CO2 adsorptions 256x178mm (96 x 96 DPI)


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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 5. Percentage of electrostatic energy the total gas-framework energy at 0.1 bar 256x178mm (96 x 96 DPI)



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 6. Spearman and Pearson correlation coefficients for each method 256x366mm (96 x 96 DPI)


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Charge Equilibration Based on Atomic Ionization in MetalâOrganic

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