Cundari Th.R. -- Computational Organometallic Chemistry-0824704789

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The Accuracy of Quantum Chemical Methods for the Calculation of Transition Metal Compounds

Michael Diedenhofen,

Thomas Wagener, and Gernot Frenking

Philipps-Universita¨t Marburg, Marburg, Germany

1. INTRODUCTION

The last decade has witnessed the establishment of quantum chemical methods as a standard tool for quantitative calculations of transition metal (TM) compounds, after numerous theoretical studies had proved that the calculated values are very accurate. The calculated data can be used to interpret experimental observations and to design new experiments and, thus, are very helpful for experimental chemistry. The theoretically predicted geometries, vibrational frequencies, bond dissociation energies, and other chemically important properties have become reliable enough to complement and sometimes even to challenge experimental data. This is particularly important for bond energies of TM compounds, which tend to be difficult to determine by experimental methods.

The situation at the end of the 1990s had dramatically changed from the 1980s, when only a small number of brave hearts of the theoretical chemistry community were tackling ‘‘the challenge of transition metals and coordination

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chemistry.’’ This was the title of a NATO Advanced Study Institute that was held in Strasbourg in 1985 (1). The proceedings of the meeting reflected the cautious and reluctant opinion of most scientists about the accuracy of quantum chemical methods that might be achieved in the field in the near future. This reservation can still be found in the foreword of the editor of the special issue of Chemical Reviews about Theoretical Chemistry that was published in 1991: ‘‘The theory of transition-metal chemistry has lagged behind the quantum theory of organic chemistry because quantitative wave functions are more complicated’’ (2).

The enormous progress in quantum chemical methods for TM compounds is due mainly to quasi-relativistic effective core potentials (ECPs) and particularly to gradient-corrected (nonlocal) density functional theory (NL-DFT), which have become standard theoretical tools in computational chemistry. Pioneering work in method development and application of DFT methods in the field of TM chemistry has been carried out by Ziegler (3). Because computational chemistry has reached a status where available methods and programs are also used by scientists who are not specialists in the field, it is reasonable to give an overview of the accuracy that can be achieved with commonly used levels of theory. This has been done by us (4) and by Cundari et al. (5) in previous reviews, which summarize theoretical studies of TM compounds with ECPs in conjunction with classical ab initio methods at the HF, MP2, and CCSD(T) levels of theory. The two reviews, which were published in 1996, also give an overview of the available ECPs that have been optimized for TM elements. The same ECPs are usually employed in DFT calculations as well, although the ECP parameters have not been optimized in the framework of DFT but rather with respect to Hartree– Fock calculations or experimental results. Available are ECPs that have been generated from atomic DFT calculations (6,7). However, calculations of a representative set of TM complexes showed that ECPs generated from HF atomic calculations may be used with little loss of accuracy in DFT calculations as well (7).

There is general agreement in the theoretical community that gradientcorrected DFT methods are in most cases superior to classical ab initio methods at the HF and MP2 levels for the calculation of TM compounds, because the accuracy of the DFT results is similar or even better than the MP2 data, while the computational costs are less. For this reason most computational chemistry groups are now using DFT methods for TM compounds. It should be noted, however, that DFT methods are inferior to high-level ab initio methods such as CCSD(T) for very accurate energy calculations. We also want to point out that the statement about the superior results of DFT methods can at present be made only for the electronic ground states of diamagnetic (closed-shell) TM compounds. Density functional theory calculations of paramagnetic TM compounds

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have been carried out (22,23), but it seems that a standard DFT method for openshell species has not yet been established.

In this chapter we want to give an overview of the scope and limitations of the presently available DFT methods commonly used for calculating TM compounds. For comparison, we also present in some cases results of ab initio calculations at the HF, MP2, and CCSD(T) levels of theory. The very large number of quantum chemical calculations of TM compounds published in the last decade makes it possible to estimate the accuracy of those DFT methods that can be considered as standard levels. The goal of this work is to serve as a guideline for nonspecialists who want to carry out DFT calculations of TM compounds. First, we will summarize the most important programs that can be employed for DFT calculations. We give an overview of the different functionals and ECPs commonly used for TM compounds. In the main part of the review we discuss selected topics and projects. These are not comprehensive but representative for the field of TM compounds. The cited references should be helpful for finding information about other fields of theoretical TM chemistry that are not discussed here.

2.QUANTUM CHEMICAL PROGRAMS, DENSITY FUNCTIONALS, ECPs, AND BASIS SETS FOR TRANSITION METALS

The most common quantum chemical programs—Gaussian (8), GAMESS (9), Turbomole (10), CADPAC (11), ACES II (12), MOLPRO (13), MOLCAS (14), and the newly developed TITAN (15)—are able to run pseudopotential calculations. Please note that CADPAC and MOLCAS can only use so-called ab initio model potentials (AIMPs) in pseudopotential calculations. Such AIMP differ from ECPs in the way that the valence orbitals of the former retain the correct nodal structure, while the lowest-lying valence orbital of an ECP is a nodeless function. Experience has shown that AIMPs do not give better results than ECPs, although the latter do not have the correct nodal behavior of the valence orbitals (16).

Most of the listed programs are also capable of running DFT calculations. In addition, there are some programs that have been developed specifically for DFT methods. The most common DFT programs are DMol (17), DGauss (18), DeMon (19), and ADF (20). The program ADF is unusual because it is the only widely distributed quantum chemical program that uses Slater orbitals as basis functions instead of the more common Gaussian functions. The use of Slater basis functions makes it a bit more difficult to compare the results of ADF with those of other programs that use Gaussian functions.

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The central question for any DFT calculation concerns the choice of the exchange and correlation functionals for the energy expression. Numerous investigations have been carried out in order to examine the reliability of different mathematical expressions for the exchange and correlation functionals, and several studies were devoted to TM compounds (3,21–24). The following conclusions can be made from these investigations and from our work that will be discussed later. First, the geometries and particularly the energies become significantly improved when nonlocal (gradient-corrected) functionals F(ρ, ρ) are employed rather than functionals that depend only on the electron densityF(ρ). Second, the nonlocal exchange functional suggested by Becke (B) in 1988 (25) has been established as a standard expression in NL-DFT calculations. Third, the choice of the best correlation functional, for which several mathematical expressions have been proposed, is less obvious than the choice of the exchange functional. The presently most popular correlation functionals are those of Perdew (P86) (26), Lee, Yang, and Parr (LYP) (27), Perdew and Wang (PW91) (28), and Vosko, Wilk, and Nuisar (VWN) (29).

The situation in choosing the proper combinations of exchange and correlation functionals became a bit confusing in the early 1990s when different functionals were combined and the resulting energy expression was given by a multiparameter fit of the functionals. The semiempirical weight factors were obtained from a fit to a set of well-established experimental values. The most commonly used functional combination of this type is the three-parameter fit of Becke (B3) (30). The original expression for the B3 hybrid functional is:

EXC 0.2(EXHF) 0.8(EXLDA) 0.72(EXB) 1.0(ECLDA) 0.81(ECNL)

A widely used variant of the B3 hybrid functional termed B3LYP (31), which is slightly different from the original formulation of Becke, employs the LYP expression for the nonlocal exchange functional ECNL. It seems that the B3LYP hybrid functional is at present the most popular DFT method for calculating TM compounds. Other widely used functionals are BP86, which gives particularly good results for vibrational frequencies (32), BPW91, and BLYP. It is a wise idea to estimate the accuracy of a functional for the particular problem at the beginning of a research project by running some test calculations before the final choice of the functionals is made. The disadvantage of DFT compared with conventional ab initio methods is that the DFT calculations cannot systematically be improved toward better results by going to a higher level of theory.

We want to point out that the development of new functionals is at present a very active field in quantum chemistry. Promising new functionals have recently been proposed by Hamprecht et al. (33) and by Becke, who introduced multiparameter fits of functionals that involve first-order and second-order density gradients (34). The limits of gradient corrections in DFT were discussed by the same author (35). The accuracy of these functionals for TM compounds has not system-

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atically been exploited yet, but it is possible that new functionals will soon be established as standard methods for TM compounds that surpass the already impressive reliability of the present methods.

The second crucial choice for a quantum chemical DFT calculation is the basis set. The valence shell of the TMs has s and d orbitals. As a minimum requirement for useful calculations it is necessary to have at least a double-zeta quality for the n(s) and (n 1)d valence orbitals. The status of the lowest-lying empty n(p) orbitals is at present controversial (36). However, it has been shown that the basis set should have at least one function that describes the empty n(p) orbital of the TM (4,37). Extra f-type polarization functions improve the accuracy particularly of the calculated energies, but it seems that they are less important for the TMs than d-polarization functions for main-group elements.

Many DFT and ab initio calculations are carried out with the frozen-core approximation for the innermost electrons, or the core electrons are replaced by pseudopotentials, mostly in the form of an ECP but sometimes as an AIMP. It is generally recognized that the outermost (n 1)s2 and (n 1)p6 core electrons should not be replaced by an ECP, but should be retained in the calculations. Small-core ECPs are more reliable than large-core ECPs, where only the (n)s and (n 1)d electrons of the TMs are calculated. Several groups developed valence basis sets in conjunction with small-core ECPs (38–41) and AIMPs (42) for the TMs. The ECP valence basis set suggested by the Stuttgart group (40) is very large and may be too big for calculations of larger molecules. It should be used for very accurate calculations. There is no report known to us that suggests that one of the other ECPs or AIMP is generally more accurate than the other.

An important theoretical aspect for calculating TM compounds concerns the effect of relativity. It is well known that relativistic effects must be considered in the calculation in order to obtain reliable geometries and energies of 2ndand 3rd-TM-row molecules (43). Elements of the first TM row are little influenced by relativity, except for copper (43,44). The most convenient way to include relativistic effects in the calculations is the use of quasi-relativistic ECPs or AIMPs. The techniques of relativistic ECPs for molecules containing transition metals and other heavy atoms have recently been reviewed (45). Most ECPs and AIMPs have been derived from scalar-relativistic atom calculations, except the ECPs for the first TM row developed by Hay and Wadt (38). Note that the spinorbit coupling term is not included in the scalar-relativistic ECPs. This seems to be not so important for the calculation of geometries, relative energies, and vibrational frequencies of closed-shell TM compounds, but spin-orbit interactions cannot be neglected for the calculations of NMR parameters of compounds of 5d TMs (see later). Various approximate treatments of relativistic effects in allelectron calculations have been suggested, and some of them have been implemented in computational chemistry programs (46). The status of relativistic DFT methods has recently been reviewed by van Wu¨llen (47). Most of the presently

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available implementations of relativistic all-electron DFT methods are based on different scalar-relativistic approximations; i.e. they are one-component approximations of the four-component Dirac equation (46,47). The results that have been published so far do not suggest that the scalar-relativistic all-electron methods are superior to quasi-relativistic ECP methods, except for the calculation of NMR parameters. The situation may become different in the future, when twocomponent methods that include spin-orbit effects become available and become more widely used for TM compounds. Work in this field is in progress (48).

3.RESULTS OF QUANTUM CHEMICAL CALCULATIONS OF TRANSITION METAL COMPOUNDS

In the following section we will discuss the results of quantum chemical calculations of TM compounds that may serve as a guideline for the search for a theoretical method. The data may also be used as an indicator of the accuracy that can be expected. The examples have been chosen to cover a large area of TM compounds that are not comprehensive, but representative of commonly used standard methods in the field.

3.1.Homoleptic Transition Metal Carbonyl Complexes

Carbonyl complexes are probably the theoretically best investigated class of TM complexes. Here we focus on the results reported in the last couple of years. Table 1 shows theoretical and experimental bond lengths and first bond dissociation energies (FBDEs) of the hexacarbonyls TM(CO)6 (TM Cr, Mo, W).

The following conclusions can be drawn from the calculated data. The DFT methods BP86 and B3LYP predict bond lengths that are in excellent agreement with experimental values that have been taken from gas-phase measurements. The calculated bond lengths at LDA are clearly inferior. The Mo–CO and W– CO bond lengths predicted at MP2 are very good, but the Cr–CO distance is too short. This is a general weakness of the MP2 method. Systematic studies have shown that MP2 gives good metal–ligand bond lengths for 4d and 5d metal, while bond lengths of 3d TMs are too short (4). The results for the FBDEs lead to a similar conclusion. BP86 and B3LYP give bond energies in good agreement with experiment. LDA and MP2 give bond energies that are in all cases too high. Previous calculations have shown that MP2 systematically overestimates the FBDE of TM complexes, particularly for the first TM row (4,49). CCSD(T) gives very accurate bond energies. Because all theoretical methods predict a higher value for the FBDE of Cr(CO)6 than the experimental value, the accuracy of the latter has been questioned (50). Please note that the methods listed in Table 1 have been used in conjunction with different basis sets, and that some results were obtained from ECP calculations while others used all-electron basis sets. However, this does not affect the general conclusions about the methods.

TABLE 1					˚													(kcal/mol) for			TM(CO)
			Bond Lengths r (A) and First TM–CO Bond Dissociation Energies D														e					6
					Cr(CO)	6							Mo(CO)	6							W(CO)		6
Method			r (TM–C)		r (C–O)	D	e	(TM–(CO))			r (TM–C)		r (C–O)	D	e	(TM–(CO))			r (TM–C)		r (C–O)		D	e	(TM–(CO))
MP2	a		1.861		1.168			58.0			2.061		1.164			46.1			2.060		1.166				54.9
		b	1.938		1.172	42.7			(45.8)	a						a									a
CCSD(T)																40.4									48.0
c			1.866		1.145			62.1			2.035		1.144			52.7			2.060		1.144				48.4
LDA
BP86		c	1.910		1.153			46.2			2.076		1.153			39.7			2.049		1.155				43.7
		d	1.921		1.155			40.7			2.068		1.155			40.1			2.078		1.156				44.8
B3LYP
Expt.		e	1.918		1.141			36.8 2			2.063		1.145			40.5 2			2.058		1.148				46.0 2
a
Ref. 100 using the standard basis set II.
b
Ref. 101.
c
Ref. 102.
d
Ref. 22.
e
Ref. 103.

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A comparison of the different methods—BP86, B3LYP, MP2, and CCSD(T)—with the same ECP/basis set combination is available from a study of the isoelectronic hexacarbonyls TMq(CO)6 with the third-TM-row elements TMq Hf 2 , Ta , W, Re , Os2 , Ir3 by Szilagyi and Frenking (51). Table 2 shows the calculated and experimental bond lengths obtained with a quasi-relativ- istic ECP and a valence basis set that has DZ P quality. Table 3 gives the FBDEs of the complexes.

The calculated bond lengths obtained via B3LYP, BP86, and MP2 are in very good agreement with the experimental results. Please note that the gas-phase

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value for the W–CO bond (2.058 A) is longer than the solid-state values (2.018–

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2.032 A). Bond lengths of donor–acceptor bonds measured in the solid state are always shorter than in the gas phase (52). This must be considered when the theoretical and experimental TM–CO distances of the TM hexacarbonyl ions shall be compared. The calculated bond energies support the conclusion that BP86 and B3LYP give values that agree with the very accurate but expensive CCSD(T) method. MP2 gives bond energies that are too high. However, the trend that is predicted for the hexacarbonyls by MP2 agrees with the other methods. Note that the BP86 and B3LYP values for W(CO)6 given in Table 3 are slightly different from the data shown in Table 1. The results were reported by different groups using different basis sets.

The vibrational spectra of TM carbonyls have also been calculated in numerous theoretical studies. Table 4 gives the theoretical and experimental stretching frequencies νCO and force constants FCO of the preceding series of isoelectronic hexacarbonyls. Figure 1 shows a plot of the t1u mode of νCO. It is obvious that the calculated trend of the force constants and vibrational frequencies is in accord with experiment. Please note that the calculations refer to harmonic fundamentals, while the experimental values are taken from the observed anharmonic modes. Systematic studies of the performance of BP86 with different ECPs for the vibrational spectra of many neutral and ionic TM carbonyls by Jonas and Thiel have shown that reliable harmonic force fields can be obtained at this level of theory (32).

Another theoretical study of TM carbonyls in which a comparison of different methods has been made was recently published by Lupinetti et al. (53). The focus of the paper was the analysis of the metal–CO bond in the series of homoleptic d 10 carbonyls TMq(CO)n, with TM Cu , Ag , Au , Zn2 , Cd2 , Hg2 , where n 1–6. In order to estimate the accuracy of the theoretical level the authors calculated the TM carbonyls of the group 11 elements Cu–Au with n 1–4 at B3LYP, BP86, MP2, and CCSD(T) levels of theory. Table 5 shows the theoretically predicted and experimental FBDEs.

The results clearly indicate a limitation of the DFT method in the calculation of the TM d10 carbonyls. The CCSD(T) values are in very good agreement with the experimental results, except for Cu(CO) and Cu(CO)2 , for which the

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		˚
TABLE 2 Calculated and Experimental Bond Lengths (A) of TM
Hexacarbonyl Complexes and CO
Compound	TM–C	C–O	Method
[Hf(CO)6]2 (Oh)	2.211	1.182	B3LYP
	2.206	1.196	BP86
	2.174(3); 2.179(3)	1.162(5); 1.165(4)	X-ray
	2.180(3)	1.162(4)
[Ta(CO)6] (Oh)	2.124	1.166	B3LYP
	2.118	1.179	BP86
	2.113	1.180	MP2
	2.083(6)	1.149(8)	X-ray
W(CO)6 (Oh)	2.074	1.151	B3LYP
	2.066	1.164	BP86
	2.060	1.166	MP2
	2.058	1.148	ED
	2.018; 2.025	1.130; 1.139	X-ray
	2.032; 2.033	1.152; 1.158
[Re(CO)6] (Oh)	2.046	1.138	B3LYP
	2.036	1.151	BP86
	2.026	1.155	MP2
	1.98(3); 2.03(6)	1.14(4); 1.12(7)	X-ray
	2.02(6); 1.89(7)	1.16(8); 1.19(8)
	2.07(7)	1.12(9)
[Os(CO)6]2 (Oh)	2.049	1.128	B3LYP
	2.038	1.141	BP86
	2.025	1.148	MP2
[Ir(CO)6]3 (Oh)	2.068	1.121	B3LYP
	2.057	1.135	BP86
	2.041	1.144	MP2
	2.05(1); 2.01(1)	1.07(1); 1.08(2)	X-ray
	2.04(1); 2.00(2)	1.12(2)
	2.02(2)
CO (C∞v)		1.150	B3LYP
		1.138	BP86
		1.152	MP2
		1.143	exptl

Source: Ref. 51.