J. Phys. Chem. 1996, 100, 8763-8769
8763
Systematic Comparison of Experimental, Quantum Mechanical, and Molecular Mechanical Bond Lengths for Organic Molecules Buyong Ma,† Jenn-Huei Lii,† Henry F. Schaefer III,‡ and Norman L. Allinger*,† Computational Center for Molecular Structure and Design and Center for Computational Quantum Chemistry, Department of Chemistry, UniVersity of Georgia, Athens, Georgia 30602-2556 ReceiVed: December 7, 1995; In Final Form: March 4, 1996X
Ab initio calculations give, with an accuracy depending on the sophistication of the method, a bond length as an equilibrium value, re. The experimental bond lengths are always vibrationally averaged and may be expressed in different ways (rg, rz, ra, etc.). Since high-quality ab initio calculations now are capable of giving bond lengths that are approximately of experimental accuracy, it is important to be able to interconvert these values. We find that the bond lengths optimized at the TZ2P+f CCSD level may be considered as the converged re values and that the MM3 and MM4 force fields successfully convert rg to re values. We also evaluated the performance of quantum mechanics at the 6-31G* MP2 and the 6-31G* B3LYP levels and found that the bond lengths (re) at the 6-31G* B3LYP level are better than these at the 6-31G* MP2 level for molecules with only first-row atoms. However, the bond lengths for the bonds involving second-row atoms are too long at the 6-31G* B3LYP level, and for these, the 6-31G* MP2 level is recommended. An empirical formula is given for the conversion of the theoretical re values calculated at these levels to the rg values.
Introduction Molecular structures may be obtained either experimentally or computationally. Since the structures determined by most kinds of experiments differ in principle from those obtained by many kinds of calculations by much more than experimental error, it is quite important to be able to interconvert these different kinds of structures. Experimental bond lengths are determined in a variety of ways, and they give rg, rz, ra, rR, or rs values normally, and the relationship between all of these values is in general understood.1 Much of this information, at the level of the quadratic approximation, has been included in the program ASYMO, which is available,2 and which will calculate these interconversions. Quantum mechanics and molecular mechanics are two popular computational methods for the study of molecular structures, and they can now achieve a precision similar to experiment. In MM33 and MM44 (and also MM1 and MM25) we used rg as our definition of “bond length”, because these are the experimental numbers of highest accuracy that are generally available for simple cases of interest. Hence these programs yield bond lengths that were decided upon as the most useful. However, quantum mechanics gives equilibrium structures (re), which are very difficult to obtain experimentally.1,6,7 Therefore, in order to make a rigorous comparison between theory and experiment, it is better to convert rg, rz, ra, or rR to the accurate ab initio value, re, or vice versa. To do so, it is necessary to take anharmonicity into account. This requires that the force field for the molecule be known. When that is available, the interconversion is straightforward.1,3,7 All of the above have been put into the MM33 and MM44 force field programs, and so, to some level of approximation, one can interconvert all of the values above. But there are a number of problems here. First, one has to be concerned about the accuracy of the re values. If one were able to do a sufficiently large ab initio calculation, one would obtain an * Address correspondence to this author. † Computational Center for Molecular Structure and Design. ‡ Center for Computational Quantum Chemistry. X Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(95)03630-6 CCC: $12.00
accurate value for re. However, real ab initio calculations always involve truncated basis sets and limited electron correlation. We can divide these approximations into two parts. First, how large must the calculation be in order to converge to the true limit? And can we do such calculations and/or reliably extrapolate calculations we can do to that limit? The second part is that for the foreseeable future, many real calculations will be done at less than that level, and therefore will have to be extrapolated to the limit. What is the best way to do that extrapolation, and what are the numbers to be used? This paper will deal with the above questions. That is, we want to be able to get from ab initio calculations, at whatever level can be used for a particular problem, the re bond lengths, estimated as accurately as possible. The other part of the problem is the conversion of the re numbers into experimental numbers. These interconversions may already be carried out by the MM3 and MM4 force field programs, so we need only discuss the accuracy of these interconversions. Theoretical Method The energy for bond stretching is normally described by a Morse function, and at either the classical or quantum mechanical level, the lowest vibrational level within that potential well relates to the experimental bond length at low temperature. As illustrated in Figure 1, the energy minimum of the well is the equilibrium position, re, that is determined by an accurate ab initio calculation. The average over the first vibrational level (properly defined) is the rg value. If the temperature of the system is raised, then higher vibrational levels are occupied in a Boltzmann distribution, and the Boltzmann average must be taken in addition. In practice, the levels are rather far apart for the stretching of ordinary bonds, and looking only at the lowest level is adequate for most purposes. The difference between rg and re then comes about because of anharmonicity. The weighted average of the bond length over the vibration is not centered at re, but at a longer distance. All of this means that re will, at least in all normal cases, be shorter than rg, and the question is by how much. It turns out that the difference is very large in the case of hydrogen, because the hydrogen has a © 1996 American Chemical Society
1.060 1.206 1.064 1.209 1.063 1.208 1.067 1.205
1.062b 1.202b
1.093 1.202 1.102 1.209 1.100 1.209 1.110 1.207
1.100a 1.203a
1.513 1.203 1.505 1.211 1.503 1.205 1.508 1.211
1.061 1.210 1.071 1.229 1.062 1.219
1.061 1.211
1.099 1.211 1.107 1.226 1.104 1.221
1.097 1.213
1.498 1.213 1.503 1.215 1.509 1.228 1.502 1.223 1.502 1.222
1.426 1.421 1.423
a Experimental values are C-H: 1.099 Å, CdO ) 1.203 Å. b Experimental values are C-H: 1.062 Å, CtC: 1.203 Å. c Reference 14. d In 10-3 Å, the rms for MM3 is 0.0055 Å and the signed average is -0.0003 Å.
-2 4
-7 -1
10 -2
1 5 1.419 1.416 1.423 1.424 1.427 1.418 1.419 1.422
1.337 1.330 1.332 1.331 1.331 1.347 1.335 1.335
1.332
1.422
0 1.332 1.329 1.336
1.330
1.527 1.523 1.524 1.532 1.523 1.530 1.530 1.524
1.429
2
-7 4 -10 0 1.525 1.094 1.521 1.085 1.522 1.090 1.535 1.091 1.534 1.089 1.532 1.090 1.530 1.095
MM3 Canh ) 2.0 Canh ) 3.0 TZ2P+f TZ2P+fd
1.523 1.087 1.526
and where T is the temperture (K), h is the Planck constant, c is the velocity of light (3.0 × 1010 cm/s), ν1 is the vibrational frequency of normal mode 1 (cm-1), k is the Boltzmann constant, L is the mass-weighted normal-mode coordinate eigenvector matrix, and δRij/δ∆Pm are the first derivatives of the interatomic distance Rij with respect to ∆Xij, ∆Yij, and ∆Zij. Here we want to emphasize the cubic anharmonicity Canh. If we expand the Morse function exactly, then Canh ) 2.0 Å-1, which is the value we used in the latest version of MM4 for purposes of interconverting different kinds of bond lengths. (We separately use 3.0 Å-1 in MM4 for purposes of calculating geometries of highly stretched bonds.) In MM3, we use Canh ) 2.55 Å-1. The results of interconversion between rg and re with the different values for Canh are reported in Table 1. Six basis sets were employed for the ab initio calculations in this study. The first, designated DZP, is the standard Huzinaga-Dunning contracted Gaussian basis8 C(9s5p/4s2p), O(9s5p/4s2p), H(9s/4s), augmented with a set of d functions on the carbon and oxygen atoms [Rd(C) ) 0.75, Rd(O) ) 0.85],
1.523 1.087
(L3(i-1)+m,1 - L3(j-1)+m,1)
m
1.530 1.087
δRij
∑ m)1δ∆P
1.531 1.096
3
GAMMA(i,j)1 )
1.523 1.089
(when T f 0 K coth (hcν1/2kT) ) 1)
TZ2P+f
hcν1 h coth 2kT 8π cν1 2
TZ2P
DELTA(T)1 )
DZP
and
6-31G**
1/2
1
6-31G*
1
molecule
1
MP2
[∑ GAMMA(i,j) DELTA(T) GAMMA(i,j) ]
TABLE 1: Effects of Basis Functions and Electron Correlation on Bond Lengths (re, Å)
[〈∆Zij2〉T]1/2 )
B3LYP 6-31G*
CCSD TZ2P+f
where 〈∆Z2〉T relates to the vibrational root-mean-square amplitude at temperature T (K).
1.526 1.093
re ) rg - 1.5Canh〈∆Z2〉T
TZ2Pc
CCSD(T)
MM3 Canh ) 2.55
very small mass and lies high up in the potential well. So here the rg value will be larger than re by something of the order of 0.02-0.03 Å. For atoms heavier than hydrogen, the differences are for the most part a good bit smaller. The heavier atom lies lower in the potential well, and the potential wells are similar in depth and shape for hydrogen and most other atoms. Still, these corrections are often an order of magnitude greater than the experimental error. In MM3 and MM4, the formula to convert the rg to the re value is
ethane C-C C-H propane C-C ethylene CdC methanol C-O acetaldehyde C-C CdO formaldehyde C-H CdO acetylene C-H CtC
MM4 Figure 1. Illustration of the definition of equilibrium (re) and thermal average value (rg) of internuclear distance.
MM4
Ma et al. MM-TZ2Pf CCSDd
8764 J. Phys. Chem., Vol. 100, No. 21, 1996
Comparison of Bond Lengths for Organic Molecules and a set of p functions on the hydrogen atoms [Rp(H) ) 0.75]. The second, designated triple zeta plus double polarization (TZ2P), is the Huzinaga-Dunning C and O (10s6p/5s3p), H(5s/ 3p) basis8 augmented with two sets of d functions [Rd(C) ) 1.50, 0.375; Rd(O) ) 1.7, 0.425] on the carbon atoms, and two sets of p functions [Rp(H) ) 1.50, 0.375] on the hydrogen atoms. The third, TZ2P+f is the TZ2P basis augmented with a set of f functions on the carbon and oxygen atoms [Rf(C) ) 0.8, and Rf(O) ) 1.4]. The fourth TZ2P+fd is the TZ2P+f basis augmented with a set of d functions on the hydrogen atoms [Rd(H) ) 0.8]. Throughout the text, five d functions and seven f functions were used. The popular 6-31G* and 6-31G** basis sets were also used. Four correlation methods were used to optimize the molecular structure: the second-order perturbation (MP2) method, the density functional theory method (B3LYP), the coupled cluster including all single and double substitutions (CCSD) method, and the CCSD with the effects of connected triple excitations added perturbatively [CCSD(T)]. In the MP2 calculations the 1s orbitals for first-row elements, and 1s, 2s, 2p orbitals for second-row elements, have been frozen. In the CCSD and CCSD(T) calculations, the 1s orbitals and the corresponding virtual orbitals are frozen. The molecular mechanics computations were carried out using the MM3 and MM4 programs. The quantum mechanical computations were carried out using the program PSI9 and the program Gaussian 92.10 Results and Discussion 1. Interconversion of ab Initio and Experimental Bond Lengths. The first question then is, how accurately can we do the ab initio calculation? What are the errors for the best calculations? Our approach to this problem has been as follows. We have selected several small molecules that contain typical bonds that are of interest. Included in this set are some single, double, and triple bonds between carbon atoms (ethane, ethylene, and acetylene) and oxygen atoms (methanol and acetaldehyde). Several ab initio calculations have been carried out for each molecule, with increasing numbers of basis functions and levels of electron correlation (Table 1). As may be seen in Table 1, the bond lengths generally shrink with increasing basis set quality but lengthen with an increasing level of correlation [from MP2 to CCSD and CCSD(T)]. The inclusion of f functions has significant effects on the bond lengths, but the hydrogen d functions have no noticeable effects. Note that the CCSD(T) bond lengths are generally 0.003-0.006 Å longer than those at the CCSD level. However, we have not correlated with 1s electrons in our calculations. Bauschlicher11 studied the effect of 1s correlation on the re value of the N2 molecule and found that it reduced the bond length by 0.002 Å. We found the similar trend for acetylene molecule. The CtC bond length is 1.206 Å at the TZ2P+fCCSD(T) with the 1s electrons correlated, while the CtC bond length is 1.209 Å at the same level without the 1s electrons correlated. Thus, as a compromise between basis functions and correlation levels, we recommended that the bond lengths at the TZ2P+f CCSD level be taken as the best theoretical re values. The above analysis is reinforced by comparisons with experimental re bond lengths for formaldehyde12 and acetylene13 (the only two organic molcules in Table 1 for which experimental re values are available). For these, the re values at the TZ2P+f CCSD level agree well with the experimental re values (Table 1). Now we are in a position to discuss the accuracy of the interconversion of the rg and re values as carried out by the MM3 and MM4 force fields. As mentioned earlier, in MM3 and MM4 we used rg as our definition of “bond length”. In
J. Phys. Chem., Vol. 100, No. 21, 1996 8765 our computations, the program first calculates the geometry which is based on the thermally averaged molecular structure (at 298 K), which gives rg values. Then the re values are calculated, based on the molecular mechanics force field and anharmonicity corrections. In Table 1, if we take the re values at the TZ2P+f CCSD level as the “converged” re values, we see that the MM3 and MM4 values agree very well with the “converged” re values. Cubic anharmonicity Canh is important for the interconversion. We note that the MM4 values with Canh ) 2.0 are the best to fit the re values at the TZ2P+f CCSD level. However, few MM4 values are currently available, as the MM4 parametrization is still in progress, but the MM3 values (with Canh ) 2.55) are acceptable. The rms deviation of MM3 and TZ2Pf CCSD bond lengths is 0.0059 Å (Table 1). 2. Evaluation of the MP2 and DFT(B3LYP) Methods. We have shown that the most reliable equilibrium molecular structures may presently be obtained at the TZ2P+f CCSD level. The agreement between experimental and TZ2P+f CCSD CdO and CtC bond lengths is within 0.001 Å. However, this level is presently impractical for routine day to day calculations. Usually, the 6-31G* MP2 level, and recently, the DFT method, are considered as good approximations. Therefore, an evaluation of the accuracy of the 6-31G* MP2 and 6-31G* B3LYP levels is certainly of importance. We have calculated the structures of 36 typical organic molecules at these two levels. The results are shown in Tables 1-5. We found that B3LYP method in general gives slightly better results than the MP2 method for the molecules containing only first-row atoms and hydrogens (Table 1). The bond lengths at the 6-31G* B3LYP level are very close to those at the TZ2P+f CCSD level. It is well-known that the MP2 methods usually underestimate the single bond and overestimate the double and triple bond length. By comparison, the B3LYP method gives reasonable bond lengths for these bonds (Tables 1, 2, and 3). However, the general trend is that the B3LYP bond lengths are slightly longer than the “converged” values at the TZ2P+f CCSD level. The rms deviation of the 6-31G* B3LYP and the TZ2P+f CCSD bond lengths is 0.0037 Å. On average, the bond lengths at the 6-31G* B3LYP level are 0.0027 Å longer than those at the TZ2P+f CCSD level. Oliphant and Bartlett14 compared the accuracy of the density functional theory (BLYP) and the coupled cluster methods and found that the bond lengths at the TZ2P BLYP level are comparable with those at the TZ2P CCSD(T) level. For the molecules with second-row atoms, the 6-31G* B3LYP method is inadequate. The bond lengths involving second-row atoms at this level are always significantly too long (Table 4). Very large basis sets are required to give reasonable values for such bond lengths. For example, even at the 6-31G(3df,2p) B3LYP level, the C-S bond lengths for dimethyl sulfide are still 0.007 Å longer than the experimental rg value (Table 5). Therefore, we recommend that 6-31G* MP2 method be used for bonds involving second-row atoms. Generally speaking, neither the 6-31G* MP2 level nor the 6-31G* B3LYP level is sufficient to give accurate re values directly. However, it is of interest to know what kinds of empirical corrections may be utilized to compare these calculations with experimental rg values. We have done the following tests: (1) directly compared the re values at the 6-31G* MP2 and 6-31G* B3LYP levels with the experimental rg values; (2) used the MM3 force field to convert the re values at the 6-31G* MP2 and 6-31G* B3LYP levels to the corresponding rg values (Rg(1) in Tables 2, 3, and 4) and then compared those with the experimental rg values; and (3) applied empirical basis set truncation and correlation corrections by the addition of a
8766 J. Phys. Chem., Vol. 100, No. 21, 1996
Ma et al.
TABLE 2: Systematic Comparison of Experimental, Quantum Mechanical, and Molecular Mechanical Bond Lengths (Å) for Hydrocarbonsa,b rg(1)
re molecule
MP2
C2H6 (ethane) C-C 1.526(-7) 1.530(3) C-H 1.093(-18) 1.096(-15) C3H8 (propane) C-C 1.526(-6) 1.532(0) C-H 1.095(-12) 1.097(-10) C4H10 (isobutane) C-C 1.529(-6) 1.535(0) 1.099(-23) 1.101(-21) C(t)-H C-H 1.096(-17) 1.097(-16) C5H12 (neopentane) C-C 1.528(-9) 1.540(3) C-H 1.095(-19) 1.097(-17) C3H6 (cyclopropane) C-C 1.503(-11) 1.509(-5) 1.085(-14) 1.087(-12) C-He C4H8 (cyclobutane) C-C 1.545(-9) 1.554(0) C-H 1.095(-14) 1.096(-13) C2H4 (ethylene) CdC 1.335(-2) 1.332(-5) C-H 1.085(-18) 1.087(-16) C3H6 (propene) CdC 1.336(-6) 1.333(-9) C-C 1.498(-8) 1.502(-4) 1.087(-17) C(sp2)-H 1.097(-20) C(sp3)-H C4H6 (1,3-butadiene) CdC 1.343(-2) 1.341(-4) C-C 1.456(-9) 1.457(-8) C-H 1.088(-20) C4H6 (dimethylacetylene) CtC 1.221(7) 1.209(-5) C-C 1.464(-4) 1.462(-6) C-H 1.094(-22) 1.097(-19) rms C-C 7.1 4.9 C-H 17.8 16.4 average C-C -5.5 -3.1 C-H -17.4 -16.0
rg(2)
MM3 re
expf rg
1.531(-2) 1.113(2)
1.522 1.090
1.533(2) 1.111(2)
1.537(5) 1.110(3)
1.534(2) 1.113(6)
1.524 1.090
1.532(3) 1.107(5)
1.534(-1) 1.112(-10) 1.110(-3)
1.540(5) 1.114(-8) 1.111(-2)
1.538(3) 1.117(-5) 1.113(0)
1.528 1.094 1.089
1.535(1) 1.122(6) 1.113(2)
1.549(12) 1.121(7)
1.532(-5) 1.109(-5)
1.544(7) 1.111(-3)
1.540(3) 1.113(-1)
1.531 1.089
1.537(3) 1.114(8)
1.512(-2) 1.518(4) 1.108(9) 1.110(11)
1.507(-7) 1.098(-1)
1.513(-1) 1.100(1)
1.512(-2) 1.503 1.514(1) 1.100e(1.087)(-1) (1.064) 1.099(2)
1.555(1) 1.118(9)
1.564(10) 1.119(10)
1.550(-4) 1.108(-1)
1.559(5) 1.109(0)
1.559(5) 1.112(3)
1.549 1.089
1.554(1) 1.109(3)
1.343(6) 1.107(4)
1.340(3) 1.109(6)
1.338(1) 1.097(-6)
1.335(-2) 1.099(-4)
1.338(1) 1.102(-1)
1.330 1.080
1.337(2) 1.103(2)
1.343(1) 1.506(0)
1.340(-2) 1.510(4) 1.110(6) 1.121(4)
1.338(-4) 1.501(-5)
1.335(-7) 1.505(-1) 1.100(-4) 1.111(-6)
1.339(-3) 1.515(9) 1.103(-1) 1.113(-4)
1.332 1.507 1.080 1.089
1.342(2) 1.506(3) 1.104(10) 1.117(8)
1.351(6) 1.465(0)
1.349(4) 1.466(1) 1.110(2)
1.346(1) 1.460(-5)
1.344(-1) 1.460(-4) 1.100(-8)
1.345(0) 1.467(2) 1.102(-6)
1.337 1.458 1.080
1.345(2) 1.465(3) 1.108(4)
1.226(12) 1.471(3) 1.118(2)
1.214(0) 1.469(1) 1.121(5)
1.221(7) 1.466(-2) 1.108(-8)
2.209(-5) 1.464(-4) 1.111(-5)
1.212(-2) 1.471(3) 1.110(-6)
1.207 1.464 1.086
1.214(1) 1.468(1) 1.116(6)
4.5 6.7
6.5 7.6
4.1 5.4
4.3 4.6
3.5 3.7
2.9 5.8
4.8 6.8
-2.0 -4.2
-0.2 -3.2
1.5 -1.2
MP2c
DFTc
1.535(2) 1.116(5)
1.539(6) 1.119(8)
1.530(-3) 1.106(-5)
1.534(1) 1.109(-2)
1.536(4) 1.118(11)
1.542(10) 1.120(13)
1.531(-1) 1.108(1)
1.539(4) 1.122(0) 1.120(7)
1.545(10) 1.124(2) 1.121(8)
1.537(0) 1.119(5)
DFT
MP2d
DFTd
rg
a C-H bond lengths are the average value. The B3LYP functionals are used in the DFT method. b The values in parentheses are the deviation: rg (computation) - rg(exp), in units of 10-3 Å. c Converted by adding the difference of MM3(rg)-MM3(re) (formula 1), see text. d Converted by formula 2, see text. e C-H bond in cyclopropane is 1.087 Å in MM3, based on the experimental rz value (1.084 Å). After the C-H bond length fixed we realized that it is inaccurate, and 1.100 Å should be used (based on the experimental rg value: 1.099 Å, ref 15), and is used here. f Reference 13. The values in parentheses are the experimental uncertainties.
constant (C) to the bond length from step 2. The bond lengths obtained in this way are designated as rg(2). The conversion formulas are
rg(1) ) re ab initio + (rg MM3 - re MM3)
(1)
rg(2) ) re ab initio + (rg MM3 - re MM3) + C
(2)
where C is estimated from the average values of the deviation between rg and rg(1) and C ) -0.010 Å for the bonds involving a hydrogen atom, when re ab initio is at either the 6-31G* MP2 or the 6-31G* B3LYP level, C ) -0.005 Å for a bond between first-row heavy atoms, when re ab initio is at either the 6-31G* MP2 or the 6-31G* B3LYP level, C ) -0.005 Å for a bond involving a second-row atom, when re ab initio is at the 6-31G* MP2 level, and C ) -0.030 Å for a bond involving a secondrow atom, when re ab initio is at the 6-31G* B3LYP level. It may be noted that the bond length rg(1) is semiempirical in nature, since the value of the anharmonicity in the vibrational amplitude is obtained from MM3 force field. However, rg(2) is more empirical than rg(1), since an empirical constant C is
included in addition. The results are reported in Tables 2, 3, and 4. Several points may be made from these comparisons: 1. As expected, C-H bonds should be corrected to account for the large anharmonicity, with the size of the correction being about 0.02 Å. 2. For aliphatic compounds (Table 2), the rg(1) bond lengths agree with experimental rg values reasonably well, indicating that both the 6-31G* MP2 and 6-31G* B3LYP methods describe hydrocarbons well enough. The rms error over rg(1) for the compounds in Table 2 is 0.0045 Å for C-C bonds and 0.0067 Å for C-H bonds at the 6-31G* MP2 level. The further empirical correction [rg(2)] improves the agreement to 0.0041 Å for C-C bonds and 0.0054 Å for C-H bonds at the 6-31G* MP2 level. Similar results are obtained at the 6-31G* B3LYP level [rms(rg(1)) is 0.0065 Å for C-C bonds and 0.0076 Å for C-H bonds, and rms(rg(2)) is 0.0043 Å for C-C bonds and 0.0046 Å for C-H bonds]. 3. For the organic molecules other than hydrocarbons (Tables 3 and 4), the empirical correction should be applied. The rg(1) bond lengths are systematically longer than experimental rg bond lengths. However, the rg(2) bond lengths, with empirical
Comparison of Bond Lengths for Organic Molecules
J. Phys. Chem., Vol. 100, No. 21, 1996 8767
TABLE 3: Systematic Comparison of Experimental, Quantum Mechanical, and Molecular Mechanical Bond Lengths for First-Row Organic Compoundsa,b rg(1)
re molecule CH3OH (methanol) C-O O-H C-H CH3CHO (acetaldehyde) CdO C-C C(sp3)-H C(sp2)-H HCOOH (formic acid) CdO C-O O-H C-H HCOHCO (trans-glyoxal) CdO C-C C-H CH3COOH (acetic acid) CdO C-O C-Ce O-H CH3COCH3 (acetone) CdO C-C C-H CH3COOCH3 (methyl acetate) C(sp2)-O C(sp3)-O C-C CdO C-H C-H C4H6O3 (acetic anhydride) C-O CdOf C-C CH3NH2 (methylamine) C-N C-H N-H CH3NHCH3 (dimethylamine) N-H C-H C-N CH3NNCH3 (azomethane) C-H C-N NdN CH3CN (acetonitrile) C-C CtN C-H CH3OCH3 (dimethyl ether) C-O C-H CH3OOCH3 (dimethyl peroxide) O-O C-O C-H CH3NHCHO (N-methylformamide) C(sp3)-H CdO C(sp3)-N C(sp2)-N C6H6 (benzene) CdC C-H C5H5N (pyridine) CdN CdC CdC C-H
rg(2)
MM3
MP2
DFT
MP2c
DFTc
MP2d
DFTd
rg
re
exp rgh
1.423(-6) 0.970(-5) 1.094(-4)
1.419(-10) 0.969(-6) 1.096(-2)
1.431(2) 0.988(13) 1.117(19)
1.427(-2) 0.987(12) 1.119(21)
1.426(-3) 0.978(3) 1.107(9)
1.422(-7) 0.977(2) 1.109(11)
1.431(2) 0.948(-27) 1.114(16)
1.423 0.930 1.091
1.429(2) 0.975(10) 1.098(1)
1.222(12) 1.502(-13) 1.109(2)
1.211(1) 1.508(-7) 1.099(-8) 1.114(-14)
1.228(18) 1.507(-8) 1.132(25)
1.217(7) 1.513(-2) 1.122(15) 1.137(9)
1.223(13) 1.502(-13) 1.122(15)
1.212(2) 1.508(-7) 1.112(5) 1.127(-1)
1.209(-1) 1.518(3) 1.107(0) 1.113(-15)
1.203 1.513 1.084 1.090
1.210(4) 1.515(4) 1.107(6) 1.128(4)
1.212(-2) 1.350(-8) 0.980(-1) 1.096(-7)
1.205(-9) 1.347(-11) 0.977(-4) 1.100(-3)
1.217(3) 1.358(0) 0.999(18) 1.119(16)
1.210(-4) 1.355(-3) 0.996(15) 1.123(20)
1.212(-2) 1.353(-5) 0.989(8) 1.109(6)
1.205(-9) 1.350(-8) 0.986(5) 1.113(10)
1.211(-3) 1.336(-22) 0.974(-7) 1.105(2)
1.206 1.328 0.955 1.082
1.214(3) 1.358(3) 0.981(3) 1.103(3)
1.220(8) 1.527(1) 1.102(-30)
1.210(-2) 1.525(-1) 1.109(-23)
1.226(14) 1.537(11) 1.127(-5)
1.216(4) 1.535(9) 1.134(2)
1.221(9) 1.532(6) 1.117(-15)
1.211(-1) 1.530(4) 1.124(-8)
1.210(-2) 1.525(-1) 1.117(-15)
1.204 1.515 1.092
1.212(2) 1.526(3) 1.132(8)
1.217(7) 1.361(0) 1.500(-17)
1.210(0) 1.359(-2) 1.508(-9) 1.093(-7)
1.223(13) 1.369(8) 1.509(-8)
1.216(6) 1.367(6) 1.517(0) 1.116(16)
1.218(8) 1.364(3) 1.504(-13)
1.211(1) 1.362(1) 1.512(-5) 1.106(6)
1.213(3) 1.366(5) 1.497(-20)e 1.109(9)
1.207 1.358 1.488 1.086
1.210(9) 1.361(9) 1.517(15) 1.100(30)
1.227(13) 1.512(-8)
1.216(2) 1.521(1) 1.095(-8)
1.233(19) 1.521(1)
1.226(8) 1.530(10) 1.121(18)
1.228(14) 1.516(-4)
1.221(7) 1.525(5) 1.111(8)
1.211(-3) 1.516(-4) 1.111(8)
1.205 1.507 1.085
1.214(4) 1.520(3) 1.103(3)
1.356(-4)
1.364(4) 1.512(16) 1.224(15)
1.363(3) 1.444(2) 1.520(24) 1.217(8) 1.113(6) 1.117(5)
1.359(-1)
1.503(7) 1.218(9)
1.355(-5) 1.436(-6) 1.511(15) 1.211(2) 1.090(-17) 1.094(-18)
1.507(11) 1.219(10)
1.358(-2) 1.439(-3) 1.515(19) 1.212(3) 1.103(-4) 1.107(-5)
1.365(5) 1.441(-1) 1.498(2) 1.215(6) 1.106(-1) 1.109(-3)
1.357 1.433 1.489 1.209 1.083 1.086
1.360(6) 1.442(6) 1.496(6) 1.209(6) 1.107(18) 1.112(18)
1.402(-3) 1.208(25) 1.501(6)
1.397(-8) 1.198(15) 1.507(12)
1.412(7) 1.213(30) 1.510(15)
1.407(2) 1.203(20) 1.516(21)
1.407(2) 1.208(25) 1.505(10)
1.402(-3) 1.218(15) 1.511(16)
1.417(12) 1.198(15)f 1.490(-5)
1.407 1.193 1.481
1.405(20) 1.183f(25) 1.495(20)
1.465(-8) 1.098(-5) 1.107(-4)
1.465(8) 1.100(-3) 1.019(-2)
1.474(1) 1.123(20) 1.037(16)
1.474(1) 1.125(22) 1.039(18)
1.469(-4) 1.113(10) 1.027(6)
1.469(-4) 1.115(12) 1.029(8)
1.463(-10) 1.120(17) 1.016(-5)
1.454 1.095 0.996
1.473(3) 1.103(3) 1.021(6)
1.108(-2) 1.098(-10) 1.458(1)
1.018(-2) 1.096(-12) 1.457(0)
1.038(18) 1.120(12) 1.466(9)
1.038(18) 1.118(10) 1.465(8)
1.028(8) 1.110(2) 1.461(4)
1.028(8) 1.108(0) 1.460(3)
1.015(-5) 1.117(9) 1.459(2)
0.995 1.095 1.451
1.020(2) 1.108(6) 1.457(6)
1.093(-12) 1.467(-15) 1.264(17)
1.096(-9) 1.468(-14) 1.244(-3)
1.462(-6) 1.180(21)
1.461(-7) 1.160(1) 1.095(-12)
1.471(3) 1.184(25)
1.470(2) 1.164(5) 1.118(11)
1.466(-2) 1.179(20)
1.465(-3) 1.159(0) 1.108(1)
1.470(2) 1.158(-1) 1.108(1)
1.461 1.154 1.085
1.468(2) 1.159(2) 1.107(4)
1.416(1) 1.099(-19)
1.410(-5) 1.103(-15)
1.424(9) 1.122(4)
1.418(3) 1.126(8)
1.419(4) 1.117(-1)
1.413(-2) 1.121(3)
1.418(3) 1.113(-5)
1.410 1.090
1.415(2) 1.118(2)
1.473(14) 1.422(0) 1.094(-7)
1.462(3) 1.415(-7) 1.097(-4)
1.483(24) 1.430(8) 1.117(16)
1.472(13) 1.423(1) 1.120(19)
1.478(19) 1.425(3) 1.107(6)
1.467(8) 1.418(-4) 1.110(9)
1.466(7) 1.424(2) 1.112(11)
1.456 1.416 1.089
1.495(12) 1.422(7) 1.101(4)
1.089(-25) 1.228(9) 1.451(-8) 1.362(-4)
1.108(-6) 1.219(0) 1.453(-6) 1.363(-3)
1.114(0) 1.234(15) 1.460(1) 1.369(3)
1.133(19) 1.225(6) 1.462(3) 1.370(4)
1.104(-10) 1.229(10) 1.455(-4) 1.364(-2)
1.123(9) 1.220(1) 1.457(-2) 1.365(-1)
1.115(1) 1.217(-2) 1.453(-6) 1.375(9)
1.090 1.211 1.444 1.368
1.114(25) 1.219(5) 1.459(6) 1.366(8)
1.397(-2) 1.087(-14)
1.397(-2) 1.087(-14)
1.407(8) 1.107(6)
1.407(8) 1.107(6)
1.402(3) 1.102(1)
1.402(3) 1.102(1)
1.397(-2) 1.105(4)
1.387 1.085
1.399(1) 1.101(5)
1.345(1) 1.396(-3) 1.394(-4) 1.088(-6)
1.339(-5) 1.396(-3) 1.394(-4) 1.088(-6)
1.349(5) 1.404(5) 1.402(4) 1.111(17)
1.345(1) 1.404(5) 1.402(4) 1.111(17)
1.344(0) 1.399(0) 1.397(-1) 1.101(7)
1.340(-4) 1.399(0) 1.397(-1) 1.101(7)
1.347(3) 1.396(-3) 1.390(-8) 1.103(9)
1.343 1.388 1.382 1.080
1.344 1.399 1.398 1.094(6)
1.105(3) 1.482(2) 1.247(3)
8768 J. Phys. Chem., Vol. 100, No. 21, 1996
Ma et al.
TABLE 3 (Continued) rg(1)
re molecule C4H4O (furan) C-O CdC C-C C-H C-H CH3F (fluoromethane) C-F C-H rms total X-Yg X-H average total X-Yg X-H
rg(2)
MM3
MP2
DFT
MP2c
DFTc
MP2d
DFTd
1.367(1) 1.366(0) 1.428(-4) 1.080 (-8) 1.081(-7)
1.364(-2) 1.361(-5) 1.436(4) 1.079(-9) 1.081(-7)
1.374(8) 1.373(7) 1.436(4) 1.102(14) 1.104(16)
1.371(5) 1.368(2) 1.444(12) 1.101(13) 1.104(16)
1.369(3) 1.368(2) 1.431(-1) 1.092(4) 1.094(6)
1.366(0) 1.363(-3) 1.439(7) 1.091(3) 1.094(6)
1.390(-1) 1.092(-16)
1.384(-7) 1.096(-12)
1.397(6) 1.115(7)
1.391(0) 1.119(11)
1.392(1) 1.105(-3)
1.386(-5) 1.109(1)
re
exp rgh
1.371(5) 1.316(-5) 1.431(-1) 1.092(4) 1.097(11)
1.364 1.354 1.423 1.070 1.074
1.366(7) 1.366(9) 1.432(2) 1.088(2) 1.088(1)
1.391(0) 1.108(0)
1.348 1.085
1.391(1) 1.108(1)
rg
9.9 8.2 12.4
8.2 6.3 10.5
12.4 10.9 15.8
11.3 7.8 14.7
7.7 8.0 7.8
6.2 5.9 6.5
7.9 5.9 10.0
-2.5 0.8 -9.5
-5.0 -3.3 -8.9
9.9 8.3 12.9
8.8 5.2 13.6
3.3 3.5 3.4
1.9 0.3 4.0
0.2 -0.3 0.7
a C-H bond lengths are the average value. The B3LYP functionals are used in the DFT method. b The values in parentheses are the deviation: rg (computation) - rg(exp), in units of 10-3 Å. c Converted by adding the difference of MM3(rg) - MM3(re) (formular 1), see text. d Converted by formular 2, see text. e Poorly determined experimentally and not included in the rms calculation. The rs bond length from microwave experiments (1.503 Å, ref 16) agrees well with our calculations. f Since the experimental uncertainty is large (0.025 Å), this value is not included in the rms calculation. g Heavy atoms only. h Reference 13. The values in parentheses are the experimental uncertainties.
TABLE 4: Systematic Comparison of Experimental, Quantum Mechanical, and Molecular Mechanical Bond Lengths for Organic Compounds Containing Second-Row Atomsa,b rg(1)
re molecule
MP2
DFT
MP2c
CH3Cl (methyl chloride) C-H 1.088(-2) 1.090(0) 1.111(21) C-Cl 1.777(-8) 1.805(20) 1.786(1) CHCl3 (chloroform) C-H 1.086(-14) 1.086(-14) 1.108(8) C-Cl 1.765(7) 1.789(31) 1.772(14) CH3SiH3 (methylsilane) C-H 1.096(0) 1.095(-1) 1.120(24) Si-H 1.490(7) C-Si 1.880(11) 1.890(21) 1.890(21) HSC2H4SH (1,2-ethanedithiol) C-H 1.094(-24) S-H 1.341(-32) 1.348(-25) 1.367(-6) C-C 1.525(-12) 1.529(-8) 1.535(-2) C-S 1.826 1.852 C-S 1.818 1.840 CH3SCH3 (dimethyl sulfide) C-S 1.807(0) 1.827(20) 1.817(10) C-H 1.093(-23) 1.095(-21) 1.117(1) CH3SSCH3 (dimethyl disulfide) 2.054(25) 2.082(53) 2.063(34) S-Se C-S 1.812(-4) 1.837(21) 1.822(6) C-H 1.091(-14) 1.095(-10) 1.114(9) C4H4S (thiophene) C-S 1.718(2) 1.736(20) 1.726(10) CdC 1.376(-3) 1.367(-12) 1.383(4) C-C 1.420(-8) 1.430(2) 1.428(0) C-H(R) 1.082(-9) 1.082(-9) 1.104(13) C-H(β) 1.085(-18) 1.085(-18) 1.108(5) rms total 13.9 17.8 11.6 7.4 25.9 9.9 X-Yg X-H 17.2 15.5 13.2 average total -5.8 0.1 8.1 -1.9 21.3 6.9 X-Yg X-H -14.0 -11.0 9.4
rg(2)
MM3
exph rg
DFTc
MP2d
DFTd
rg
1.113(23) 1.814(29)
1.101(11) 1.781(-4)
1.103(13) 1.784(-1)
1.105(15) 1.788(3)
1.082 1.090(2) 1.779 1.785(1)
1.108(8) 1.796(38)
1.098(-2) 1.767(9)
1.098(-2) 1.766(8)
1.099(-1) 1.755(-3)
1.077 1.100(5) 1.748 1.758(2)
1.119(23) 1.520(37) 1.900(31)
1.110(14) 1.885(16)
1.109(13) 1.510(27) 1.870(1)
1.112(16) 1.483(0) 1.875(7)
1.088 1.096(1) 1.453 1.483(1) 1.866 1.869(1)
1.118(0) 1.374(1) 1.539(2)
1.357(-16) 1.530(-7)
1.108(-10) 1.364(-9) 1.534(-3)
1.114(-4) 1.343(-30) 1.540(3) 1.853 1.835
1.090 1.317 1.530 1.842 1.824
1.837(30) 1.119(3)
1.812(5) 1.107(-9)
1.807(0) 1.109(-7)
1.808(1) 1.112(-4)
1.798 1.807(2) 1.088 1.116(3)
2.091(62) 1.847(31) 1.118(13)
1.817(1) 1.104(-1)
1.817(1) 1.108(3)
2.031(2) 1.819(3) 1.111(6)
2.022 2.029(3) 1.809 1.816(3) 1.088 1.105(5)
1.744(28) 1.374(-5) 1.438(10) 1.104(13) 1.108(5)
1.721(5) 1.378(-1) 1.423(-5) 1.094(3) 1.098(-5)
1.714(-2) 1.369(-10) 1.433(5) 1.094(3) 1.098(-5)
1.722(6) 1.370(-9) 1.423(-5) 1.100(9) 1.101(-2)
1.714 1.363 1.415 1.078 1.078
21.6 25.8 16.9
8.4 7.4 9.3
9.1 4.9 11.6
9.7 5.4 12.4
16.7 21.3 12.6
0.7 1.9 -0.6
1.2 -0.3 2.6
0.5 0.4 0.5
re
1.118(11) 1.373(15) 1.537(6) 1.824(2)
1.716f 1.379f 1.428f 1.091f 1.103f
a C-H bond lengths are the average value. The B3LYP functionals are used in the DFT method. b The values in parentheses are the deviation: rg (computation) - rg(exp), in units of 10-3 Å. c Converted by adding the difference of MM3(rg) - MM3(re) (formula 1), see text. d Converted by formula 2, see text. e The bond between two second row atoms is not included in rms calculation. f Obtained by adding MM3 value of (rg - rR) to experimental rR values (C-S 1.716 Å, CdC 1.378 Å, C-C 1.427 Å, C-H(R) 1.069 Å, and C-H(β) 1.081 Å). g Heavy atoms only. h Reference 13. The values in parentheses are the experimental uncertainties.
correction applied, agree with experimental rg values reasonably well. For X-H and X-Y type bonds, where X may be C, O,
N, S, or Cl, only one example was studied where both X and Y were S, and one example was studied where both X and Y were
Comparison of Bond Lengths for Organic Molecules TABLE 5: Basis Set Dependence of the C-S Bond Lengths (re) of Dimethyl Sulfide (Experimental rg ) 1.807 Å) basis set
MP2 (Å)
B3LYP (Å)
6-31G* 6-31G** 6-311G* 6-311G** 6-31+G(3df,2p)
1.807 1.805 1.804 1.802
1.827 1.826 1.825 1.826 1.814
O, so that we are uncertain of the accuracy of rg(2) when X and Y are both electronegative or second-row atoms, and such cases need further study. It should be noted that the accuracy of the experimental numbers is limited as well. The uncertainties of experimental values are frequently reported, and some uncertainties are included in Table 2. However, we should be cautious in comparing experimental uncertainties with the deviations between empirical and experimental values. For example, Kuchitsu has pointed out that notwithstanding all possible caution in estimating a “reasonable” uncertainty in each of the geometric parameters obtained by electron diffraction, they can never be immune from hidden systematic errors.17 Good electron diffraction structures of quite small or symmetrical molecules frequently yield bond lengths between firstrow atoms from carbon to fluorine with an accuracy of 0.0020.004 Å. When there are similar but nonequivalent bonds in a molecule that are not resolved in the bond-length part of the radial distribution function (the usual case), their average can be measured with similar accuracy, but the individual values are frequently off by far more than this. The present work shows that the ab initio and density functional methods at the 6-31G* MP2 and 6-31G* B3LYP levels, with appropriate corrections for truncations, can yield an accuracy that is at least close to competitive with experiment for bonds between this same group of atoms. Molecular mechanics calculations can suffer from at least two possible general defects. One of these is improperly chosen parameters. The second is the assumption of the transferability of parameters between molecules. The present work again shows that the choice of parameters for MM3 reproduces the experimental/ab initio data reasonably well, indicating that neither of these is a serious problem in the cases examined. Additional errors may be anticipated in general for bonds involving conjugated systems, involving hydrogen, involving atoms to the left of carbon in the periodic table, or involving atoms which are in the second row, or lower, in the periodic table. Conclusions We find that the bond lengths for a group of representative molecules at the TZ2P+f CCSD level maybe considered as the converged re values as a compromise between basis set and electron correlation limits. The agreement between experimental and TZ2P+f CCSD CdO and CtC re bond lengths is within 0.001 Å. The MM3 and MM4 force fields successfully calculate and convert the rg values to re values for several organic
J. Phys. Chem., Vol. 100, No. 21, 1996 8769 molecules (ethane, ethylene, acetylene, methanol, formaldehyde, and acetaldehyde) with a rms deviation of 0.0059 Å for this set (Table 1). We also evaluated the performance of quantum mechanics at the 6-31G* MP2 and the 6-31G* B3LYP levels and found that the bond lengths (re) at the 6-31G* B3LYP level are calculated slightly more accurately than those at the 6-31G* MP2 level for molecules containing only first-row atoms and hydrogen. The rms deviation from the experimental bond lengths is 0.0099 Å for the 6-31G* MP2 level and 0.0082 Å for the 6-31G* B3LYP level (Table 3). However, the bond lengths for bonds involving second-row atoms are significantly too long at the 6-31G* B3LYP level, and the 6-31G* MP2 level is recommended for calculations on such bonds. A calculational method was devised to convert theoretical re values to rg values. Acknowledgment. This research was supported by Tripos Associates, 1699 South Hanley Road, St. Louis, MO 63144. References and Notes (1) Kuchitsu, K.; Cyvin, S. J. In Molecular Structures and Vibrations; Cyvin, S. J., Elsevier Publishing Co.: New York, 1972. (2) Available from the Quantum Chemistry Program Exchange (ref 3). Hedberg, L.; Mills, I. M. J. Mol. Spectrosc. 1993, 160, 117. (3) Allinger, N. L. Yuh, Y. H.; Lii, J.-H. Molecular Mechanics. The MM3 Force Field for Hydrocarbons I, II, and III. J. Am. Chem. Soc. 1989, 111, 8551-8582, and subsequent papers. The MM3 program is available to commercial users only from Tripos Associates, 1699 South Hanley Road, St. Louis, MO 63144, and to academic users only from the Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN 47405. The MM4 program will be available shortly from the same sources. (4) Allinger, N. L.; Chen, K.; Lii, J.-H. J. Comput. Chem. 1996, 17, 642. (5) Allinger, N. L. J. Am. Chem. Soc. 1977, 99, 8127. (6) Nakata, M.; Kuchitsu, K. J. Mol. Struct. 1994, 320, 179. (7) Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules; D. Van Nostrand: Princeton, NJ, 1945. (8) (a) Huzinaga, S. J. Chem. Phys. 1965, 42, 1293. (b) Dunning, T. H. J. Chem. Phys. 1970, 53, 2823. (c) Dunning, T. H.; Hay, P. J. In Modern Theoretical Chemistry; Schaefer, H. F., Ed.; Plenum Press: New York, 1977; Vol 3, pp 1-27. (9) Janssen, C. L.; Seidl, E. T.; Hamilton, T. P.; Yamaguchi, Y.; Remington, R. B.; Xie, Y.; Vacek, G.; Sherrill, C. D.; Crawford, T. D.; Fermann, J. T.; Allen, W. D.; Brooks, B. R.; Fitzgerald, G. B.; Fox, D. J.; Gaw, J. F.; Handy, N. C.; Laidig, W. D.; Lee, T. J.; Pitzer, R. M.; Rice, J. E.; Saxe, P.; Scheiner, A. C.; Schaefer, H. F. PSI2. 0.8; A program to do ab initio computations with SCF and post-SCF methods (configuration interaction and coupled cluster methods); available from Psitech, Inc., Watkinsville, GA, 30677, 1994. (10) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M.; Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A.; Head-Gordon, M.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Stewart, J. J. P.; Pople, J. A. Gaussian 92/DFT, Revision F.2; Gaussian Inc., Pittsburgh, PA, 1993. (11) Bauschlicher, C. W.; Partridge, H. J. Chem. Phys. 1994, 100, 4329. (12) Yamada, K.; Nakagawa, T.; Kuchitsu, K.; Morino, Y. J. Mol. Spectrosc. 1971, 38, 70. (13) All experimental data, except indicated otherwise, are cited from: Structure Data of Polyatomic Molecules; Kuchitsu, K., Ed.; SpringerVerlag: Berlin, 1992. (14) Oliphant, N.; Bartlett, R. J. J. Chem. Phys. 1994, 100, 6550. (15) Yamamoto, S.; Nakata, M.; Fukuyama, T.; Kuchitsu, K. J. Phys. Chem. 1985, 89, 3298. (16) Van Eijck, B. P.; Van Zoeren, E. J. Mol. Spectrosc. 1985, 111, 138. (17) Kuchitsu, K. In Structure Data of Polyatomic Molecules; Kuchitsu, K., Ed.; Springer-Verlag: Berlin, 1992.
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