# Quantum Monte Carlo study of porphyrin transition metal complexes.

**ABSTRACT** Diffusion quantum Monte Carlo (DMC) calculations for transition metal (M) porphyrin complexes (MPo, M=Ni,Cu,Zn) are reported. We calculate the binding energies of the transition metal atoms to the porphin molecule. Our DMC results are in reasonable agreement with those obtained from density functional theory calculations using the B3LYP hybrid exchange-correlation functional. Our study shows that such calculations are feasible with the DMC method.

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**ABSTRACT:**A projector Monte Carlo method based on Slater determinants (PMC-SD) is expanded to excited state calculations. Target excited states are calculated state-by-state by eliminating the components of the lower states from the imaginary-time propagator. Test calculations are performed for the singlet excited states of H2O and LiF. As the calculations of H2O show, the accuracy of the PMC-SD method is improved systematically by increasing the number of walkers. The full-CI energies are obtainable as a limit for a given basis set. The avoided crossing of covalent and ionic states in the dissociation of LiF is well reproduced using the PMC-SD method.Chemical Physics Letters 01/2010; 485:367-370. · 2.15 Impact Factor

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Quantum Monte Carlo study of porphyrin transition metal complexes

Jun Koseki,1Ryo Maezono,2,3,a?Masanori Tachikawa,1,3M. D. Towler,4and R. J. Needs4

1Quantum Chemistry Division, Graduate School of Science, Yokohama City University, Seto 22-2,

Kanazawa-ku, Yokohama 236-0027, Japan

2Japan Advanced Institute of Science and Technology, School of Information Science, Asahidai 1-1, Nomi,

Ishikawa 923-1292, Japan

3Precursory Research for Embryonic Science and Technology, Japan Science and Technology Agency,

Kawaguchi, Saitama, Japan

4TCM Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge,

CB3 0HE, United Kingdom

?Received 20 May 2008; accepted 9 July 2008; published online 29 August 2008?

Diffusion quantum Monte Carlo ?DMC? calculations for transition metal ?M? porphyrin complexes

?MPo, M=Ni,Cu,Zn? are reported. We calculate the binding energies of the transition metal atoms

to the porphin molecule. Our DMC results are in reasonable agreement with those obtained from

density functional theory calculations using the B3LYP hybrid exchange-correlation functional. Our

study shows that such calculations are feasible with the DMC method. © 2008 American Institute

of Physics. ?DOI: 10.1063/1.2966003?

I. INTRODUCTION

Transition metal ions in biological molecules play a very

important role in the field of bioinorganic chemistry1and

organometallic chemistry.2Representative roles include the

transport of oxygen,3electron transfer,4and the catalytic na-

ture of metalloproteins,5which are essential processes in

metabolism.6Electronic structure calculations of such sys-

tems are becoming more important in bio-based research

fields, such as “drug design” in pharmaceutics.

An accurate treatment of electron correlation in these

systems with standard quantum chemistry techniques re-

quires the use of multireference methods. Traditional quan-

tum chemical multireference methods are not currently com-

putationally feasible for such systems, which has stimulated

interest in the density functional theory ?DFT? approach.7

Although DFT is a powerful method for treating electronic

correlations with a reasonable computational cost, its predic-

tions often depend significantly on the specific choice of the

exchange-correlation density functional.7Calibrating such

inconsistencies is one of the missions of the quantum Monte

Carlo ?QMC? method, which is a more reliable tool for cal-

culating the energies of correlated electron systems.

It is vital to use pseudopotentials ?also known as effec-

tive core potentials or ECPs? in QMC calculations for atoms

with large atomic numbers Z because the computational cost

is estimated to scale as Z5.5–6.5.8,9Pseudopotentials which are

well suited for use in QMC calculations have recently been

developed.10,11Nonlocal exchange and correlation effects are

very important for transition metal ions. It is therefore of

technical interest to examine QMC pseudopotential calcula-

tions for systems including transition metal ions.

The first difficulty we face is the preparation of suitable

basis sets for calculating the single particle orbitals used in

the trial wave functions. Gaussian basis sets are commonly

used in biosystems, and popular ones include the Hay–Wadt

?LANL1DZ, LANL2DZ? basis sets, which are designed for

use with pseudopotentials.12However, the pseudopotentials

commonly used in quantum chemistry calculations are not

particularly suitable for QMC calculations because they di-

verge at the ionic centers. Such divergences lead to poor

behavior in the local energy unless the proper cusp

condition13is applied. Orbitals expanded in a Gaussian basis

set cannot satisfy the cusp condition, and the local energy

then diverges at the ionic center and fluctuates wildly nearby,

which leads to biases and even instabilities in diffusion quan-

tum Monte Carlo ?DMC? calculations. Although the cusp

conditions at the ionic center may be enforced by modifying

the Jastrow factor or correcting the orbitals,14better results

may be obtained by using pseudopotentials which are finite

at the ionic center, such as those of Refs. 10 or 15. However,

standard basis sets are not provided with these pseudopoten-

tials, and therefore we have devised a simple scheme for

generating Gaussian basis sets for them. We start with a stan-

dard Gaussian basis set for the Hay–Wadt pseudopotentials,

and then optimize it for each molecule or atom by minimiz-

ing the energy within Hartree–Fock ?HF? self-consistent field

?SCF? calculations.

Porphyrin metal complexes are typical prosthetic groups

of metalloproteins,16

such as the reaction centers of

hemoglobin,17vitamin B12,18etc. Electronic structure calcu-

lations for these complexes have been reported before.19–22

From the viewpoint of biochemistry, the existence of side

chains is important, as it gives many possible combinations

of side chain and transition metal ion ?M? at the porphin

center. Nevertheless, in the present study, we treat the por-

phyrin metal complex ?MPo? without side chains, so that we

can concentrate on the pseudopotential issues and avoid the

complicated structural optimizations required for low-

symmetry systems. Taking NiPo, CuPo, and ZnPo as ex-

amples, we evaluate the binding energy ?E of the transition

metal ion to the MPo. QMC calculations of electronic exci-

a?Electronic mail: rmaezono@mac.com.

THE JOURNAL OF CHEMICAL PHYSICS 129, 085103 ?2008?

0021-9606/2008/129?8?/085103/5/$23.00 © 2008 American Institute of Physics

129, 085103-1

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Page 2

tations of the porphyrin molecule were reported by Aspuru-

Guzik et al.,23who obtained excellent agreement with ex-

periment, but we are not aware of any previous QMC

calculations on MPo.

In this paper, we have calculated porphyrin metal com-

plexes using the DMC method. The plan of this paper is as

follows. In Sec. II, we describe the systems studied, and in

Sec. III we describe the pseudopotentials and basis sets used

in the calculations. The variational Monte Carlo ?VMC? and

DMC methods and calculations are described in Sec. IV. Af-

ter giving the definition of the binding energy used here in

Sec. V, the results of the calculations are reported and com-

pared in Sec. VI. We draw our conclusions in Sec. VII.

II. STRUCTURES OF THE PORPHYRIN METAL

COMPLEXES

The general porphyrin metal complex we have consid-

ered is shown in Fig. 1. The “FreeBase” ?porphin without the

two inner hydrogen atoms? and MPo structures were opti-

mized within D4hpoint group symmetry by performing all-

electron ?AE? B3LYP ?Ref. 24? DFT calculations using

6-31G** Gaussian basis sets. Calculations of the vibrational

frequencies of the molecules confirmed that the optimized

structures for CuPo and ZnPo are stable. B3LYP and MP2

calculations gave an unstable vibrational mode for NiPo, and

following the instability we obtained a structure of C1sym-

metry. We investigated this distortion further using the more

sophisticatedcomplete-active-space

?CASSCF? method, in which all possible configurations in an

self-consistent-field

active space ?Nocc.,Nvir.? consisting of Nocc.occupied and

Nvir.virtual orbitals are included. When we increased the

active space from ?2,2? to ?6,6? ?Ref. 25? the equilibrium

structure returned to D4hsymmetry, showing that the C1dis-

tortion is an artifact.20We therefore employed the D4hstruc-

ture for NiPo as well. The bond lengths of the optimized

structures are shown in Table I. All calculations were carried

out using the

abinitio

GAUSSIAN03.26

quantumchemicalpackage

III. PSEUDOPOTENTIALS AND BASIS SETS

We performed AE and pseudopotential calculations at

the HFSCF and B3LYP levels, and QMC calculations using

pseudopotentials. We used both large-core ?Ar core? and

small core ?Ne core? pseudopotentials for the M. We have

investigated the effects of varying the parameters of the cal-

culations, including the size of the basis set. These results

enable us to investigate the results obtained with different

pseudopotentials, explore the convergence with respect to

basis set size and study the differences obtained at different

levels of theory.

We usedthe following

/pseudopotentials and basis sets:

combinationsofAE

• AE; 6-31G**, cc-pvTZ?-NR?,

• small core; LANL2DZ, CRENBL, SBKJC-VDZ,

• large core; LANL1DZ, CRENBS, TN?pp.

The cc-pvTZ ?Ref. 27? ?correlation consistent polarized

valence triple zeta? basis sets are larger than the 6-31G**

ones. LANL1DZ ?LANL2DZ? denotes the Hay–Wadt large

?small? core pseudopotentials for the transition metal ions

with LANL1DZ ?LANL2DZ? ?=Hay-Wadt? basis sets.14

CRENBL ?CRENBS? ?Ref. 28? is a small ?large? shape con-

sistent pseudopotential which includes an averaged treatment

of relativistic effects, and for which basis sets are provided.

SBKJC-VDZ ?Ref. 29? is a relativistic small core pseudopo-

tential with basis sets supplied. TN?pp stands for the smooth

large core HF pseudopotential of Trail and Needs.10For the

TN?pp, we used optimized LANL1DZ basis sets, and for the

lighter atoms described by TN?pp pseudopotentials we used

optimized 6-31G** basis sets. We optimized the exponents

and contraction coefficients of the basis sets by approxi-

mately minimizing the HF-SCF energy. Tests have indicated

that HFSCF orbitals are near optimal for sp atoms,30,31al-

TABLE I. Optimized bond lengths in angstrom for each complex using B3LYP/6-31G**. The definitions of the

bond lengths are shown in Fig. 1. FreeBase does not contain a transition metal atom, and in this case r1

corresponds to half the distance between two N atoms on opposite sides of the ring.

NiPo CuPoZnPoFreeBase

r1

r2

r3

r4

r5

r6

r7

?M–N?

?N–C?

?C–C?

?C–C?

?C–C?

?C–H?

?C–H?

1.957

1.381

1.438

1.358

1.380

1.082

1.085

2.007

1.376

1.443

1.361

1.389

1.082

1.085

2.043

1.374

1.445

1.364

1.396

1.082

1.086

2.066

1.355

1.461

1.362

1.409

1.083

1.090

FIG. 1. Structure of the porphyrin metal complex, where M=Ni,Cu,Zn.

085103-2Koseki et al.J. Chem. Phys. 129, 085103 ?2008?

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Page 3

though they may not be as accurate for transition metal at-

oms. The basis sets optimizations were carried out separately

for each molecule and atom considered. QMC calculations

were performed using the TN?pp for both the transition metal

ion and FreeBase, with optimized LANL1DZ basis sets.

IV. VMC AND DMC METHODS

The VMC energy is evaluated as the expectation value

of the Hamiltonian Hˆwith a many-body trial wave function

?,

E =??*Hˆ?dR

??*?dR

=????2?−1Hˆ?dR

????2dR

,

?1?

where R is the 3N-dimensional vector of the electron posi-

tions, and the energy has been written as an average of the

“local energy” EL=?−1Hˆ? over the probability distribution

p?R?=???2/????2dR. The energy expectation value is evalu-

ated by Monte Carlo integration, using the Metropolis algo-

rithm to generate electronic configurations distributed ac-

cording to p?R?. The statistical efficiency of the Monte Carlo

integration improves as the quality of ? improves because

EL=?−1Hˆ? becomes a smoother function of R.

All of the QMC calculations were performed with the

CASINO QMC code.32We first performed VMC calcula-

tions using a trial wave function consisting of the product of

up and down-spin determinants of the HFSCF orbitals,

??R? = D↑?R?D↓?R?.

When using the pseudopotentials which diverge at the ion

center, we have forced the molecular orbitals to satisfy the

proper cusp condition at each ionic center using the proce-

dure introduced by Ma et al.14The radial part of each mo-

lecular orbital is replaced by a form with the required cusp

inside some small radius around the ionic center and

smoothly connected to the Gaussian orbitals outside. The

cusp corrected molecular orbitals lower the VMC energy and

reduce the standard error in the energy by a factor of about

10. The cusp correction procedure is not required for the

TN-pp pseudopotentials, and the calculations with these

pseudopotentials reproduce the HFSCF results to within sta-

tistical error bars.

In the VMC calculations with Slater–Jastrow wave func-

tions, the trial functions took the form

?2?

??R? = exp?J?R??D↑?R?D↓?R?,

where exp?J?R?? is a Jastrow correlation factor. Where nec-

essary, we used the cusp corrected HFSCF orbitals described

above to form the determinants D↑and D↓. The Jastrow

factors took the form33

J?R? =?

i?j

i

?3?

u?rij? +?

I?

?I?riI? +?

I?

i?j

fI?riI,rjI,rij?, ?4?

where i and j denote electrons and I denotes ions. The u term

describes homogeneous, isotropic, electron-electron correla-

tions, the ? term one-body isotropic electron-nucleus corre-

lations, and the f term isotropic electron-electron-nucleus

correlations. The terms are represented as power series in

their arguments, constrained to enforce the electron-electron

cusp conditions while maintaining the electron-nucleus cusp

conditions. The coefficients in the power expansions are de-

termined by minimizing the self-consistent unreweighted

variance of the energy using a VMC procedure. As the coef-

ficients appear linearly in the Jastrow factor, the optimization

can be performed efficiently using the scheme devised by

Drummond et al.34

In the DMC method, the ground-state component of a

trial wave function is projected out by evolving an ensemble

ofelectronic configurations

Schrödinger equation. Attempts to carry out this procedure

exactly result in a “fermion sign problem,” which is removed

by constraining the nodal surface of the wave function ?the

surface in configuration space on which the wave function is

zero and across which it changes sign? to equal that of the

trial wave function. The DMC energy calculated with this

fixed-node constraint is higher than the exact ground-state

energy, and becomes equal to it when the fixed nodal surface

is exact.

We used the optimized Slater–Jastrow wave functions as

trial functions for the DMC calculations. Time steps of 0.01

and 0.001 a.u. were used for the DMC calculations. In the

atomic DMC calculations, we used an average population

size of 1000 configurations, while in the MPo and Po calcu-

lations we used an average population size of 8000 configu-

rations, which are expected to lead to negligible population

control errors. Furthermore, population control errors tend to

cancel in energy differences because they always increase

the energy.35,36

usingthe imaginary-time

V. DEFINITION OF THE BINDING ENERGY

We evaluate the energy difference between the neutral

states,

?E = ?E?Free Base? + E?M?? − E?MPo?,

?5?

which requires the energies of FreeBase ?E?FreeBase??, the

neutral atomic energy for the M ?E?M??, and the energy of

MPo ?E?MPo??. We can use this quantity to compare the

results of the pseudopotential and AE calculations. We note

that the quantity

?E?Po2−? + E?M2+?? − E?MPo??6?

is often studied in experimental syntheses in aqueous solu-

tion because the ionic states, X2?, are much more stable than

the neutral states due to the solvation by water molecules.

Since we have assumed isolated MPo systems, we have not

evaluated this quantity here.

VI. RESULTS

The calculated binding energies ?E for NiPo, CuPo, and

ZnPo are given in Tables II–IV, respectively. We found that

the HFSCF results for CuPo were sensitive to the initial

guess, and we experimented with a number of initial guesses

in order to achieve satisfactory results. The differences

between the AE results obtained with the 6-31G** and

cc-pvTZ-NR transition metal basis sets are as large as

0.13 a.u., and we conclude that the 6-31G**transition metal

085103-3 Quantum Monte Carlo study of metal complexesJ. Chem. Phys. 129, 085103 ?2008?

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Page 4

basis set is inadequate for these calculations. The differences

between the AE results obtained with the 6-31G** and

cc-pvTZ FreeBase basis sets are 0.02 a.u. or less, and we

conclude that the 6-31G**basis is adequate for the AE C, N,

and H atoms. The pseudopotential results do not depend

strongly on the choice of basis set, except for the curious

case of CuPo with the small core LANL pseudopotential and

6-31G** FreeBase basis set within B3LYP. With the excep-

tion of this case, our results provide good evidence that the

pseudopotential binding energies with the larger basis sets

are well converged. The DMC results do not appear to be

very sensitive to the basis set used.

The larger basis set AE results are in reasonable agree-

ment with the small core pseudopotential results. The results

with small core pseudopotentials are in fairly good agree-

ment with one another, except for the exceptional CuPo case.

There is a tendency for the large core pseudopotential bind-

ing energies of NiPo and CuPo to be somewhat smaller than

the small core ones. Presumably this is due to the additional

approximation of treating the 3s and 3p electrons within the

pseudopotential. The agreement between the various large

core results is not as good as for the small core ones, indi-

cating sensitivity to the form of the pseudopotential.

We found the DMC calculations to be unstable with the

LANL pseudopotentials which diverge at the origin, but they

were stable with the smooth TN-pp pseudopotentials. Due to

the large computational cost of the calculations, the statisti-

cal error bars obtained with a time step of 0.001 a.u. are

considerably larger than those obtained with a time step of

0.01 a.u. The energies obtained with the two time steps are

within statistical errors of one another, indicating that the

time step errors are not larger than the error bars. The DMC

binding energy of CuPo is larger than the HFSCF and

B3LYP values obtained with the same pseudopotentials, in-

TABLE II. Comparison of ?E ?atomic units? from different methods for

NiPo.

Transition metal FreeBaseMethod

?E

AE

AE

AE

AE

AE

AE

Small core

Small core

Small core

Small core

Small core

Small core

Small core

Small core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

6-31G**

cc-pvTZ-NR

cc-pvTZ-NR

6-31G**

cc-pvTZ-NR

cc-pvTZ-NR

LANL2DZ

LANL2DZ

CRENBL

SBKJC VDZ

LANL2DZ

LANL2DZ

CRENBL

SBKJC VDZ

LANL1DZ

CRENBS

TN?pp

TN?pp

LANL1DZ

CRENBS

TN?pp

TN?pp

TN?pp

TN?pp

TN?pp

6-31G**

6-31G**

cc-pvTZ

6-31G**

6-31G**

cc-pvTZ

cc-pvTZ

6-31G**

6-31G**

6-31G**

cc-pvTZ

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

TN?pp

6-31G**

6-31G**

6-31G**

TN?pp

6-31G**

TN?pp

TN?pp

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

HFSCF

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

B3LYP

HFSCF

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

B3LYP

DMC ??t=0.01?

DMC ??t=0.001?

DMC ??t=0.01?

0.301

0.310

0.309

0.406

0.321

0.341

0.307

0.305

0.307

0.311

0.338

0.314

0.306

0.320

0.283

0.442

0.225

0.226

0.243

0.266

0.271

0.292

0.375?2?

0.32?2?

0.333?1?

TABLE III. Comparison of ?E ?atomic units? from different methods for

CuPo.

Transition metalFreeBase Method

?E

AE

AE

AE

AE

AE

AE

Small core

Small core

Small core

Small core

Small core

Small core

Small core

Small core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

6-31G**

cc-pvTZ-NR

cc-pvTZ-NR

6-31G**

cc-pvTZ-NR

cc-pvTZ-NR

LANL2DZ

LANL2DZ

CRENBL

SBKJC VDZ

LANL2DZ

LANL2DZ

CRENBL

SBKJC VDZ

LANL1DZ

CRENBS

TN?pp

TN?pp

LANL1DZ

CRENBS

TN?pp

TN?pp

TN?pp

TN?pp

TN?pp

6-31G**

6-31G**

cc-pvTZ

6-31G**

6-31G**

cc-pvTZ

cc-pvTZ

6-31G**

6-31G**

6-31G**

cc-pvTZ

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

TN?pp

6-31G**

6-31G**

6-31G**

TN?pp

6-31G**

TN?pp

TN?pp

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

HFSCF

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

B3LYP

HFSCF

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

B3LYP

DMC ??t=0.01?

DMC ??t=0.001?

DMC ??t=0.01?

0.356

0.321

0.320

0.399

0.269

0.289

0.311

0.310

0.311

0.318

0.286

0.529

0.257

0.269

0.298

0.228

0.293

0.293

0.196

0.162

0.240

0.255

0.249?2?

0.237?7?

0.230?1?

TABLE IV. Comparison of ?E ?atomic units? from different methods for

ZuPo.

Transition metal FreeBaseMethod

?E

AE

AE

AE

AE

AE

AE

Small core

Small core

Small core

Small core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

Large core

6-31G**

cc-pvTZ-NR

cc-pvTZ-NR

6-31G**

cc-pvTZ-NR

cc-pvTZ-NR

CRENBL

SBKJC VDZ

CRENBL

SBKJC VDZ

LANL1DZ

CRENBS

TN?pp

TN?pp

LANL1DZ

CRENBS

TN?pp

TN?pp

TN?pp

TN?pp

TN?pp

6-31G**

6-31G**

cc-pvTZ

6-31G**

6-31G**

cc-pvTZ

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

6-31G**

TN?pp

6-31G**

6-31G**

6-31G**

TN?pp

6-31G**

TN?pp

TN?pp

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

HFSCF

HFSCF

B3LYP

B3LYP

HFSCF

HFSCF

HFSCF

HFSCF

B3LYP

B3LYP

B3LYP

B3LYP

DMC ??t=0.01?

DMC ??t=0.001?

DMC ??t=0.01?

0.307

0.274

0.272

0.316

0.237

0.257

0.252

0.266

0.207

0.229

0.272

0.232

0.237

0.237

0.240

0.198

0.252

0.273

0.262?2?

0.266?8?

0.265?1?

085103-4Koseki et al. J. Chem. Phys. 129, 085103 ?2008?

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Page 5

dicating the importance of an accurate description of valence

correlation for this system. The DMC binding energies of

NiPo and ZnPo are similar to the HFSCF and B3LYP values

obtained using the same pseudopotentials.

Liao and Scheiner23have reported DFT results for MPo

structures with four phenyl side chains and for the same

model MPo system that we have used. Our geometries are in

good agreement with theirs, with a maximum deviation in

M–N bond length of 1.1% for CuPo. They used a frozen

core approximation, which is roughly equivalent to using

small core pseudopotentials, and so our LANL small core

binding energies should be comparable with theirs. We find

quite good agreement with our B3LYP-DFT results, with a

maximum deviation in the binding energy of 0.032 a.u. for

ZnPo.

VII. CONCLUSION

We have performed calculations for transition metal at-

oms, porphin and porphyrin M complexes using a variety of

methods. AE and pseudopotential methods were used, in-

cluding both small and large core transition metal pseudopo-

tentials. DMC calculations were only feasible with large core

transition metal pseudopotentials. The DMC calculations

with pseudopotentials that diverge at the origin were un-

stable, but they were stable when the nondivergent TN-pp

pseudopotentials were used. We calculated the binding en-

ergy of the transition metal ion to the porphin. There are

significant variations between the binding energies calculated

with the B3LYP, HFSCF, and DMC methods, and with the

different AE and pseudopotential treatments. The results are

insensitive to whether the C, N and H atoms are treated with

AE or pseudopotential methods, but they depend signifi-

cantly on the treatment of the M. Comparisons between our

small and large core results and the DFT calculations of Liao

and Scheiner23suggest that small core M pseudopotentials

are required to obtain accurate results, particularly for CuPo

and NiPo. We found reasonable agreement between our DFT

binding energies and those of Liao and Scheiner.23Our study

has demonstrated that DMC calculations for MPo systems

are possible.

ACKNOWLEDGMENTS

We thank Professor Umpei Nagashima for help and for

useful discussions. Financial support was provided by Pre-

cursory Research for Embryonic Science and Technology,

Japan Science and Technology Agency ?PRESTO-JST? for

R.M. and M.T., by a Grant-in-Aid for Scientific Research in

Priority Areas Development of New Quantum Simulators

and Quantum Design ?No. 17064016? of The Japanese Min-

istry of Education, Culture, Sports, Science, and Technology

?KAKENHI-MEXT? for R.M., and by the Engineering and

Physical Sciences Research Council ?EPSRC? of the United

Kingdom for M.D.T and R.J.N. Our calculations were per-

formed mainly using the Hitachi SR11000 computer at the

High Performance Computing System of Hokkaido Univer-

sity and the National Institute for Materials Science ?NIMS,

Tsukuba Japan?, and Opteron AMD clusters at Yokohama

City University and at NIMS. The authors would also like to

thank Professor Teruo Matsuzawa ?JAIST? for his generous

provision of computing facilities at JAIST.

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