J . Phys. Chem. 1994$98, 3967-3971
3967
Density Functional Study of the Structures and Nonlinear Optical Properties of Urea? David A. Dixon. DuPont Central Research and Development, Experimental Station, P.O. Box 80328, Wilmington, Delaware 19880-0328
Nobuyuki Matsuzawa' SONY Corporation Research Center, 174 Fujitsuka-cho, Hodogaya- ku, Yokohama 240, Japan Received: November 5, 1993; In Final Form: January 24, 1994"
The geometry, conformations, and vibrational spectra of urea have been predicted at the density functional theory level (DFT).The molecule is predicted to have a lowest energy C2 conformer but with very floppy NH2 groups. The torsion barrier about the C-N bond is predicted to be about 7 kcal/mol. Calculations on model compounds were also performed in order to better understand the dependence of the structural predictions on the level of DFT employed. Thedipole moments, polarizabilities, and first- and second-order hyperpolarizabilities of urea were calculated and found to be in good agreement with the available experimental values. The nonlinear optical properties of the small molecules H2, N2, CO, CHI, NH3, H20, HF, and formaldehyde have also been calculated including gradient (nonlocal) corrections. This shows that gradient corrections make only a small contribution to these properties and that the inclusion of diffuse functions is more important.
1. Introduction
Urea is one of the prototypical molecules used in the study of the nonlinear optical (NLO) properties of molecular systems. It is also an important component of many biological systems (e.g., sulfonylureas). Thus, its structure and conformational behavior are of interest. The urea molecule contains two amide functions, and it is important to understand the interactions of these two units. For example, there has been significant debate as to whether formamide is planar or nonplanar.' Thus an important question to ask is what is the expected structure if two amides are in conjugation as found with urea? The structure of urea has been extensively studied in the crystal and is planar in this environment.2 The microwave structure of urea in the gas suggests a structure with less than C, ~ y m m e t r y .A ~ low-energy mode of 227 cm-l has been observed by matrix isolation spectroscopy, and this was interpreted as being consistent with a nonplanar str~cture.~There have been three recent ab initio molecular orbital predictions of the molecular structure of the isolated urea m o l e ~ u l e .Meier ~~ and Coussens reported HF/6-3 lG* and MP-2/6-3 lG* geometry optimizations and found the C2 conformer to be more stable than the Cz, structure by 1.6 kcal/mol at the H F Ievel and 2.6 kcal/ mol at the MP-2 level.5d However, they did not characterize their structure as stationary points. Kontoyianni and BowenSc optimized the geometry of urea also at the HF/6-31G* level and subsequently studied the rotation barrier about the C-N bond at this level. They obtained a rotation barrier of 17.7 kcal/mol at the HF/6-31G* level and 16.0 kcal/mol at the MP-2/6-31G* level for a rigid rotation. Relaxation of the geometries leads to a barrier of -8 kcal/mol. Again, the structures were not characterized as being stationary points. Gobbi and FrenkingSC calculated various conformations of urea at the HF/6-3 lG* and MP-2/6-31G* levels and characterized the stationary points by calculating second derivatives to obtain vibrational frequencies. Planar C, urea has two imaginary frequencies at both levels. Improving the computational level to MP-4/6-3 1 1G** raises the energy of the planar structure to 3.5 kcal/mol above that of the C2 form. At the MP-2/6-31G* level, the rotation barrier is 8.1 kcal/mol, and this decreases to 7.4 kcal/mol when the difference in zero-point energies (AZPE) are included. The AZPE term is f
0
Contribution no. 6727. Abstract published in Aduance ACS Abstracts, March 1, 1994.
0022-3654/94/2098-3967$04.50/0
even larger for the energy difference between the C2 and C, structures reducing the energy by 1.4 kcal/mol at the MP-2/ 6-31G* level from 2.6 kcal/mol to yield 1.2 kcal/mol. We are interested in investigating new theoretical/computational methodologies for obtaining the proper descriptions of molecular electronic structure that are computationally very efficient, yet give good accuracy. In a formal sense, the easiest way to do this is to solve the Schrodinger equation for electronic motion with the most extensive one-particle basis sets and the most extensive treatment of the correlation energy or n-particle problem. However, these ab initio molecular orbital theory methods do not scale well with the size of the molecular system. For N basis functions, the Hartree-Fock method scales formally as N4, and if correlation effects are included, the scaling goes as Nm,m 1 5 . Although many molecular chemical problems can be solved at the Hartree-Fock level, the inclusion of correlation effects is often required for quantitative predictions. We are particularly interested in using computational methods that scale with smaller values of m, yet still provide quantitative results. The method that we have been testing for chemical systems along with other research groups is that of density functional theory (DFT) in both the local (LDFT) and nonlocal (NLDFT) approximations.6 We have had a significant amount of success in using LDFTfor predicting thestructures andvibrational spectra of molecular systems.6fqgJ We have also been using DFT methods to predict polarizabilities (a)and first- (8) and second-order (y) hyperpolarizabilities. We first applied LDFT to the prediction of a and y for benzene and C60, and found reasonable values.8a Subsequently, we calculated the values for 1.1 (dipole moment), a,0, and y for a number of substituted benzenes and model polyenese8b To obtain good agreement with experiment, it was necessary to add diffuse functions to the basis set. For this, we followed the workof Guan et al.,9awho showed that thevalue of a and j3 for simple molecules (H2, N2, CO, CH4, NH3, H20, and HF) at the LDFT level could be substantially improved if field-induced polarization (FIP) functions9 (which are quite diffuse) were added to the basis set. We also included nonlocal corrections in our study and found that this led to modest improvements in the calculated values. Handy and co-workers10 have recently reported LDFT values for a and j3 for CH2O and CH3CN. At the LDFT level with a TZ2P +diffuse basis set, reasonable agreement with experiment is found. 0 1994 American Chemical Society
3968 The Journal of Physical Chemistry, Vol. 98, No. 15, 1994
TABLE 1: Mesh Parameters parameter A
R ,
(au)
thresholdb L a x X”.
@
FINE” 1.2 12.0 0.000 01 29 302
XFINE” 1.5 15.0
0.000 001 35 434
Mesh parameter in DMol. Angular sampling threshold.
Below we describe DFT calculations at the local and nonlocal level for urea. The conformational analysis of urea is described as well as the gas-phase vibrational frequencies. We show that both a C, and a C2 structure can be formed with the C2 structure being the global minimum. Calculations of the NLO properties of urea at the LDFT and NLDFT levels with FIP functions are presented as well as for the fragment molecules CH2O and NH3. For completeness, we also present NLDFT/FIP calculations on the molecules studied by Guan et al.: H2, N2, CO, CH4, H20, and HF.98
Dixon and Matsuzawa The program DGauss employs Gaussian basis functionsinstead of numerical functions. Calculations were done with all electron basi~sets~~oftheform (41/1)forHand s: (7111/411/1)for C, N, and 0. This corresponds to a polarized triple-[valence basis set for the heavy atoms and a polarized double-[ basis set for H. The fitting basis sets for the electron density and the exchangecorrelation potential have the forms [4/1] for Hand [8/4/4] for C, N, and 0. The calculations were done at the self-consistent gradient corrected (nonlocal) level with the nonlocal exchange potential of Becke15 together with the nonlocal correlation functional of Perdew.Is The local potential fit of Vosko et aI.19 was used. The geometries were optimized by analytic gradient methods.I1J2,20 The calculation of the nonlinear optical properties was done with a finite field approach2’ by using numerical basis sets and DMol. In this approach, the response of the ground-state charge distribution to an applied electric field at zero frequency is calculated. A molecule in an applied electric field will exhibit an induced dipole moment which can be expanded in a Taylor series in powers of the applied electric field as shown in eq 2.
2. Calculations
The calculations described below were done with the program systems DMolIl and DGauss.12 For the DMol calculations, the atomic basis functions are given numerically on an atom-centered, spherical-polar mesh. The radial portion of the grid is obtained from the solution of the atomic DFT equations by numerical methods. Because the basis sets are numerical, the various integrals arising from the expression for the energy need to be evaluated over a grid in terms of radial functions and spherical harmonics. The number of radial points NR(out to the maximum distance, R,,,) is given as
where Z is the atomic number. The angular integration points Ne are generated at the NR radial points to form shells around each nucleus. The value of Ne ranges from 14 to Nm depending on the behavior of the density and a maximum 1 value for the spherical harmonic, Lax. These quantities for the two meshes used in this study are given in Table 1. The Coulomb potential corresponding to the electron repulsion term is determined directly from the electron density by solving Poisson’s equation. The form for the exchange-correlation energy of the uniform electron gas for the LDFT calculations is that derived by von Barth and Hedin (BH).I3 Calculations were also performed including gradient (nonlocal) corrections (NLDFT). The gradient correction to thecorrelation potential is that derived by Lee, Yang, and Parr (LYP)I4 and to the exchange potential is that derived by BeckeIs (BLYP). The LYP correction also requires the use of an appropriate local potential as described in ref 14. Geometries for the calculations of the NLO properties were optimized by using analytic gradient methods at the LDFT level with the von Barth and Hedin potential and a double numerical basis set augmented by polarization functions (DNP).11J3J6The density was converged to 1V for the geometry optimizations, and the FINE mesh was used. For the calculations of the NLO properties, we began with the DNP basis set. Additional calculations were done with a larger basis set obtained by augmenting the DNP basis set with the FIP basis functions given by Guan et Calculations were performed with FIP s and p functionsg for first-row atoms and FIP p function on H (DNP+spd+p). The exponents of the tight d polarization were changed to the values recommended by Guan et aL9 This basis set is denoted as DNP+2. The multipolar fitting functions for the model density used to fit the effective potential have angular momentum numbers, I, one greater than that of the polarization functions.
A similar expression is available for the energy. A dipole expansion as opposed to an energy expansion was used because the field occurs as the third power in the expression for y based on the dipole expansion and the fourth power in the energy expansion. The dipole expansion leads to fewer numerical precision problems. Of course, DFT methods cannot be used predict general excited states, so the sum-over-state methods21 as implemented by others in semiempirical methods as a single excitation configuration interaction calculation cannot be performed. To use the perturbation expansion, the value of the applied field must be small enough for the expansion given in eq 2 to be valid but large enough to incorporate higher order nonlinearities; thus, a value for the field of 0.005 au was usually used. The expansion in eq 2 was truncated after the term in y. We have shown previouslys that a value of 0.005 au for the electric field yields converged @ and y values for a number of benzene derivativesat this computational level. Because of the small values of the field, the evaluation of y requires 2-3 orders of magnitude more accuracy in the energy expansion as compared to the dipole expansion. The minimal number of points required to evaluate eq 2 was used following the work of Kurtz.22b Although one could use a more extensive set of field strengths to evaluate eq 2, this would become prohibitively expensive for the large molecules that weintend tostudywith thismethod. Theequations given by Sim et a1.2t for p , a,8, and y in terms of dipole moments calculated in an applied electric field wereused. Thescalar values for p , a,8, and y can be calculated from their vector or tensor components as follows:22b
a = &/3
(4)
I
Y = C~ttjj15
(6)
iJ
where 8, in eq 5 is defined as
(7) For the calculation of hyperpolarizabilities, the XFINE mesh with a density convergence of 1o-S was used.
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 3969
Structures and Optical Properties of Urea H
H2
'.
2$
o=c
H2
\
7.""'H' 7'"' \ 0-c
\
0-c,
0 ° F
/
H-N
\H
H'
C3"
CAa)
-
\
cs
D3h
Ci@)
C,(C)
Figure 1. Calculated conformers of urea, NH3, and formamide. For
formamide, structures Cl(b) and Cl(c) have C, symmetry.
3. Results and Discussions 3.1. Conformational Analysis of Urea. For urea, the energies and structures of the five conformers, CZ,C,(a), Cb,C,(b), and C,(c), shown in Figure 1 werecalculated, at the BH/DNP, BLYP/ DNP, and BP/TZVP levels. The calculated energies and vibrational frequencies are shown in Tables 2 and 3, respectively, together with available experimental values.50+ The C2,structure corresponds to planar urea, and the DFT calculations at both the local and nonlocal levels show that the structure is not a minimum or a transition state as it is characterized by two imaginary frequencies (Table 3). These frequencies correspond to the inphase and out-of-phase combinations of the NH2 inversion motions. The in-phase combination leads to the C,(a) structure and the out-of-phase combination leads to the C2 structure. Both of these structures are minima with the C2 structure being the global minimum. The CS(a)structure lies about 1 kcal/mol above the Czstructure for all three DFT calculations. The C2, structure lies above the C2 structure by 1.41-3.24 kcal/mol depending on the computational level. This variation in energy between the numerical and gaussian basis set results is somewhat larger than might be expected. If zero-point energy (ZPE) differences are included, there is a significant change in the relative energies. Because the energy of the inversion modes is not negligible, the energy of the Cz, structure is significantly reduced. At the BH/DNP level, the Cb and C,(a) structures are of comparable energy, both just less than 1 kcal/mol above the Cz structure. At the BLYP/DNP level, the C2, structure is now only 1.6 kcal/mol above the CZ structure. At the BP/TZVP level, the CzUstructure is essentially isoenergetic with the CZstructure and is even slightly lower in energy. This suggests that the potential for inversion at the nitrogens is not well described by a harmonic potential and that this approximation is failing in urea. This has previously been suggested for formamide where inversion motion at nitrogen may actually follow a quartic as opposed to a quadratic motion.' These results can be summarized by stating that the inversion motions at nitrogen in urea will have large amplitudes.
-
The C,(b) structure is calculated to be 8 kcal/mol above the C2structure, which reduces to 7 kcal/mol if the ZPE correction is included. At the BH/DNP and BLYP/DNP levels, two imaginary frequencies are calculated for the structure, whereas at the BP/TZVPlevel, only one imaginary frequency is calculated. We note that Gobbi et al. reported only one imaginary frequency for the C,(b) structure calculated a t the MP-2/6-31G* level.sc For the Cs(c) structure, again two imaginary frequencies are calculated at the BH/DNP and BLYP/DNP levels. This structure is higher in energy by -5.5-6.0 kcal/mol as compared to the C,(b) structure. Our calculated energy differences can be compared to those reported earlier. The Hartree-Fock value is similar to the BP/ TZVP value, the MP-2/6-31G* value is similar to the BH/DNP value, and the MP-4/6-31 l G * value is similar to the BLYP/ DNP value for the relative energy of the CZ,conformer. For the C,(b) conformer, the trend is different. The Hartree-Fockvalue of 9.0 kcal/mol is similar to the BH/DNP value, the MP-2/631G* value is similar to BLYP/DNP value, and the MP-4/6311G** value is similar to the BP/TZVP value. To better understand the conformational behavior of urea at the various computational levels, we calculated the inversion barriers of NH3 and formamide and the rotation barrier about the C-N bond in the latter. The calculated energies of the C3, and D3h conformers of NH3 (Figure 1) are given in Table 4. Calculated vibrational frequencies aregiven in Table 5. The D3h conformer is calculated to have one imaginary frequency, showing that this conformer is a transition state. The conformer is 7.2 kcal/mol above the C3, conformer at the BH/DNP level, which increases to 7.9 kcal/mol at the BLYP/DNP level. Inclusion of the zero-point energy reduces the barrier to 6.8 kcal/mol at the BLYP/DNP level,which is larger than the corresponding potential barrier in urea by more than a factor of 4. The inversion barrier at the BP/TZVP level is calculated to be 4.73 kcal/mol uncorrected for zero-point effects and 3.75 kcal/mol when they are included. The experimental value is 5.8 kcal/m01.~~Thus the numerical basis sets overestimate the barrier height, whereas the BP/TZVP calculations underestimate the inversion barrier making planar structures at nitrogen more probable. The calculated energies and vibrational frequenciesz4 of formamide are also shown in Tables 4 and 5, respectively. The conformers we calculated are shown in Figure 1 (Cl,C,(a), C,(b), and C,(c) conformers). The results on vibrational frequencies show no negative frequencies for the C1 conformer, whereas one imaginary frequency was calculated for the other conformers with the DNP basis sets. This shows that the C1 conformer is the minimum, and the other conformers are transition states with this basis set. The inversion barrier of the N H z group is significantly smaller than that for urea and NH3, and actually if we include zero-point energy, the C,(a) structure has the lowest energy. At the BP/TZVP level, formamide is planar even without zero-point corrections. The relative energies of the C,(b) and Cs(c) conformers are essentially identical, in contrast to urea, and the C-N rotational barrier in formamide is significantly higher than that in urea even for the C,(c) structure. On the basis of the results for NH3 and formamide, the BP/ TZVP calculations apparently underestimate the inversion barrier at nitrogen. This could be the reason why the C h structure has the lowest energy if the ZPE is included at this level for urea and also why the C,(b) and C,(c) structures are true transition states for torsion about the C-N bond in urea at the BP/TZVP level. The results all suggest that urea will be nonplanar with a C2 structure in the low-temperature gas with very floppy NH2 groups. The calculated geometry parameters of urea are listed in Table 6 together with experimental values.2csc The geometry of urea is best known from the crystal. In the crystal, the molecule is planar with C2, symmetry (Table 6). Considering the presence
3970 The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 TABLE 2 conformer
c2
Dixon and Matsuzawa
Calculated Energies of the Conformers of Urea in kcal/mol** HF/Id MP-2/Id HF/” HF/” MP-2/IC MP-4/” 6-31G*
6-31G*
6-31G*
6-31G*
0.0
0.0
6-31G*
0.0
0.0
0.0
6-31 1G**
0.0
1.6
2.6
1.6
1.7 (0.2) 9.0
2.6 (1.1) 8.1 (7.4)
3.5 7.6
CS(d
c2a
Cdb)
-8
CSW
BH/DNP
BLYP/DNP
BP/TZVP
0.00
0.00 1.22 (0.73) 3.24 (1.64) 8.29 (7.16) 13.55 (11.99)
0.00 0.94 (0.38) 1.41 (-0.09) 7.88 (7.12) 13.90 (12.88)
1.15 (0.82) 2.54 (0.92) 8.67 (7.59) 14.28 (12.83)
@Thecalculated energy for the C2 conformer at the BH/DNP level, BLYP/DNP, and BP/TZVP levels is -223.622 258 8 au, -225.298 645 6 and -225.374 121 au, respectively. Numbers in parentheses include zero-point correction.
TABLE 3: Calculated Vibrational Frequencies of the Conformers of Urea in cm-l c2 CS(a) c2u BH/DNP BLYP/DNP BP/TZVP expt4vb BH/DNP BLYP/DNP BP/TZVP BH/DNP BLYP/DNP BP/TZVP a” 278 257 184 (34) a2 4861’ 506i 421i 383 357 (67) 227 a1 365 445 409 (320) b2 4561 3361 443 401i 462 440 (109) 410? a’ a2 447 488 410 (70) 361 309 (0) a“ 495 a2 326 477 453 (2.6) al 452 515 463 (32) 484 542? a‘ 498 a1 455 460 (3.7) 539 528 (82) a2 540 537 569 539 (58) 599 565 (214) 578? a’ 531 (7.3) 555 b2 535 a2 567 575 545 (25) 559 618? a” 536 (13) 558 bl 563 677 572 (67) al 622 736 (20) 755 790? a’ 737 (3.0) b2 742 738 780 774 (68) 763 a2 778 927 (9.4) a1 904 922 899 922 (7.6) 938 (8.6) a’ 952 922 al 948 975 (25) 941 (15) 1014 bl 974 1001 945 034 1003 (27) a2 1023 a“ 986 1161 1131 (3.2) a1 1091 097 (0.1) a’ 1138 1068 170 1136 (2.7) al 1144 1344 1374 (211) bl 1366 1403 346 1371 (209) 1394 380 (221) a” 1397 a2 1398 1590 1565 (201) bi 1564 556 (275) a” 1544 1539 606 1577 (180) 1594 a? 1553 1544 1610 1579(14)‘ a; 1571 1560 (0.2) 1577(0.9j 1594 a’ 1558 al 1559 1607 a‘ 1766 1689 1728(422) al 1756 1678 1723 (464) 1738(406) 1734 al 1777 1712 3440 a” 3410 3376 3474(36) bi 3453 3432 3506 (55) 3368 3470(37) a2 3415 a‘ 3419 3386 3485(7) a1 3462 3442 3516 (4.5) 3476(2.6) 3440 al 3418 3371 3541 3497 3607(41) bl 3594 3591 (34) 3548c a” 3580 3650 (50) a2 3533 3479 3548 a’ 3542 3497 3609(30) al 3598 3582 3652 (45) 3480 3591 (27) al 3533 Cdb) CdC) BH/DNP BLYP/DNP BP/TZVP BH/DNP BLYP/DNP BP/TZVP 313i a” 400i 3571 351i a” 3541’ 270i a” 127i 198i 236 (250) a” 209i 3341 215 (196) a’ 457 48 1 457 (1 3) 448 458 (5.6) a’ 462 539 (24) a” 525 545 532 (5.2) a’ 533 541 588 (1.5) a’ 585 571 570 (0.8) a’’ 589 589 669 657 (1.6) a” 669 663 a“ 676 675 (7.9) 847 (64) a’ 785 840 747 (137) a’ 892 831 925 (63) a’ 879 a‘ 942 966 889 853 (9.6) 1056 (4.2) a’ 1054 1042 1054 (13) a’ 1056 1070 a” 1157 1233 1191 (0.1) 1187 1146 (0.3) a’’ 1183 a’ 1277 1290 1311 (178) 1234 1251 (231) a’ 1347 1534 (124) a‘ 1508 1548 a’ 1511 1548 1549 (81) 1587 (67) a’ 1572 1600 1567 (79) a’ 1570 1637 a’ 1767 1697 1729 (416) a’ 1790 1700 1751 (385) a’ 3346 3293 3377 (1 1) 3290 3416 (5.4) a’ 3336 a” 3406 3360 3442 (13) a’ 3426 3374 3493 (21) a’ 3444 3422 3493 (40) a” 3446 3408 3507 (1 1) a’ 3600 3568 3639 (51) a’ 3579 3559 3639 (29) Values in parentheses for BP/TZVP are infrared intensities in km/mol. ,?” are those given in ref 4. Value in ref 4 is 3448 cm-I. Probable typographical error. @
of hydrogen bonding in the crystal and the low inversion frequency, it is not surprising that urea is planar in thecrystal. TheNLDFT bond lengths are longer than the LDFT values for bonds between heavy atoms, and shorter for bonds involving hydrogen. The C=O bond is calculated to be too short, whereas the C-N bonds are longer than the experimental values. The geometry does depend on the conformation. In the Cb structure where the “classical resonance” effect should be strongest, the C=O bond has its longest value and the C-N bond its shortest as would be expected. For the CZ,structure, the C = O bond is 0.002-0.004 A shorter than the experimental value, and the C-N bond is 0.04-0.06 A longer than the experimentalvalues. The calculated bond angles are in somewhat better agreement differing by 1-2O from the experimentalvalues. The LDFT values are in agreement with the MP-2/6-3 1G* values reported previously.5d In Table 7,we show the calculated rotational constants at the BP/TZVP level as compared to the experimental values. From
these values, it is clear that the molecule will be nonplanar. We then used the calculated bond angles, the calculated values for r(N-H) and fit r(C-0) and r(C-N). Reasonable agreement with the experimental rotational constants can be obtained. The value for r(C-0) obtained in this manner is about 0.03 A shorter than the calculated value for the C2 structure and 0.02 A for the C, and C, structures. Ther(C-N) distance is 0.03-0.04 A longer than the calculated value as expected from comparison to the X-ray data. As the DFT values for r(N-H) may be slightly long, these values were varied by -0.01 and -0.02 A but as shown in Table 7, the rotational constants are only weakly dependent on r(N-H). It is useful to note that the rotational constants a t this level do not allow us to choose between the C2 and CJa) structures. The differences in the calculated ab initio molecular orbital and DFT values and the experimental values are far outside normal differences. This suggests that it would be useful to obtain a gas-phase structure of urea as well as to study its geometry with
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 3971
Structures and Optical Properties of Urea
TABLE 4 Calculated Energies of the Conformers of NHs and Formamide BH/DNP
BLYP/DNP
BP/TZVP
0.00 4.73 (3.75)
NHsb
c3u
0.00
D3d
7.24 (6.07)
0.00 7.92 (6.84)
0.00 0.07 (-0.30) 18.46 (18.03) 20.06 (19.45)
0.00 0.25 (-0.37) 16.55 (15.84) 18.21 (17.37)
formamide' CI CAa) Cdb) CdC)
a Numbers in parentheses correspond to energies with zero-point correction. The calculated energies for the C3uconformer of NH3 are -56.1 13 943 34 au. -56.553 322 82 and -56.585 095 au at the BH/ DNP, BLYP/DNP, and BP/TZVP levels, respectively. The calculated energy for the C1 conformer of formamide is -168.647 723 9 and -169.918 915 3 au at the BH/DNP and BLYP/DNP levels.
TABLE 5 Calculated Vibrational Frequencies of NHh Formamide, and Formaldehyde in cm-l BH/ DNP
BLYP/ BH/ BLYP/ DNP DNP DNP
BP/ TZVP
BPI TZVP
EXPTa
~~
NH3 C3" 1128 (a) 1645 (e) 3317 (a) 3450 (e)
D3r
1169 1678 3281 3403
c3v
8871 9421 968 (173) 1492 1530 1624 (41) 3462 3457 3401 (0.8) 3684 3668 3539 (3.4)
D3r c 3 v --
9241' 1455 (87) 3561 (0) 3769 (89)
950 (a) 1627 (e) 3337 (a) 3769 (e)
Formamide Cda) 297 535 622 672 1031 1256 1336 1520 1778 2853 3396 3551
371 549 642 986 1031 1252 1358 1577 1701 2865 3384 3527
206i 534 641 972 1011 1249 1335 1524 1772 2847 3405 3561
Ci(b)b 5661' 584 832 912 1093 1190 1317 1565 1768 2884 3294 3365
5361 565 853 904 1049 1225 1326 1593 1719 2906 3268 3336
CI(C)b 5121 5151' 607 594 808 794 918 875 1054 988 1190 1198 1312 1373 1574 1651 1807 1748 2789 2802 3295 3248 3370 3341
1107 1188 1436 1780 2776 2829
1109 1166 1436 1745 2776 2834
310i 535 627 976 1009 1223 1351 1564 1699 2866 3405 3556
C, 146(232)(a") 536(9.4) (a') 623 (15) (a") 979 (3.2) (a") 1008 (3.0)(a') 1232(91) (a') 1357(4.6)(a') 1546(60)(a') 1736(386)(a') 2864(94) (a') 3469 (26) (a') 3617(38) (a')
289 (a") 565 (a') 602 (a") 1030 (a') 1059 (a") 1255 (a') 1378 (a') 1572 (a') 1734 (a') 2852 (a') 3451 (a') 3545 (a')
Formaldehyde 1167 (bz) 1249 (bl) 1500 (al) 1746 (al) 2783 (al) 2843 (bl)
a Frequencies for CH2O and NH3 from ref 25. Frequencies for formamide from ref 24. Actual symmetry is C,.
higher levels of theory (more complete treatments of the oneparticle and n-particle spaces). In the geometries for the transition state for rotation about the C-N bond, the calculated C-N2 bonds are significantly longer than for the other conformers, consistent with a loss of conjugation. The largest difference in bond angles between the two conformers is found for ~ ( N I - C - N ~ ) .The value for the C,(c) conformer is larger by 4.5' at the BH/DNP level; the B(C-NI-H1) angle also
increases by 2.3'. The N2 atom is less pyramidal for the C,(c) conformer, as the sum of the angles of B(C-Nz-H3), B(C-N2H3') and B(H3-NrHi) is 325' and 313.9' for the C,(c) and C,(b) conformers, respectively. The vibrational fundamentals for urea have been partially assigned based on Ar matrix isolation spectra. The assignments based on our calculations are reasonable (there is a likely typographical error in Table 4 of ref 4, where the b2 stretch at 3448 cm-I should be 3548 cm-I). A most interesting feature in the spectra is the possible presence of two species. Rather than assigning them to dimers, it is possible that the C, and C2 isomers are actually being observed. The presence of a very low inversion mode for urea at 227 cm-1 is not consistent with any of our calculated transitions for the C2 structure. As discussed above, it is possible that the inversion motion is very anharmonic, which would lower this frequency. It is useful to note that the lowest inversion frequency for the C2 structure is much higher than the lowest one for the C, structure. The lowest inversion frequency for the C, structure is not inconsistent with the experimental value of 227 cm-l. The geometry parameters for the fragments, NH3, formaldehyde, and formamide are listed in Table 8, together with the experimental values.1d.26 We have also shown in Table 5 the calculated vibrational frequencies of the molecule^.^^.^^ The predicted geometry for formaldehyde is in reasonable agreement with the experimental values with the C=O bond being 0.010.02 A too long. Again, the NLDFT value is longer than the LDFT value. This effect is in contrast to what is found in urea, where the calculated value for the C=O bond is shorter than found previously. As found for urea and in other studies, the bonds involving hydrogen are long by 0.02-0.025 A. Similarly, the calculated N-H bond in NH3 is longer than the experimental value, and the HNH bond angle is almost 2' too large. For formamide, the Cl(a) and C, structures are essentially isoenergetic. The sum of the bond angles around the nitrogen atom are 354.3' (Cl(a)) and 360' (C,) at the BH/DNP level. This shows that the NH2 group in the Cl(a) structure is more planar than that in urea (C2 conformer) and NH3, where the sums are 340.3' and 313.2', respectively. The geometries of the molecules, Ha, N2, CO, CH4, H2O and HF are given in Table 7 together with experimental and the calculated vibrational frequencies in Table 9. The calculated geometry parameters are in reasonable agreement with the experimental values. For the diatomic molecules, H2, N2, and CO, the calculated bond lengths are 0.028,0.019, and 0.017 A longer than the measured ones, respectively, at the BH/DNP level. For CHI and H20, the same trend is found with differences of 0.009 and 0.028 A, respectively. The calculated value for the bond angle in H2O is 1.Oo larger than the experimental value as found for NH3. The calculated frequencies are less than the experimental v a l ~ e s consistent ~ ~ . ~ ~ with the results of other workers.30 3.2. Hyperpolarizabilities. Hyperpolarizabilities of Urea and Formaldehyde. The values of p, a,6, and y for formaldehyde are given in Table 10 together with other calculated values? and those for the five conformers of urea are given in Table 11. (See Figure 2 for the molecular orientations with respect to Cartesian axes.) The value of p for formaldehyde is not strongly dependent on the basis set or level of calculation, whereas the value of a increases by about 10% if diffuse functions and gradient corrections are used. The value of 6 does depend somewhat on the field strength and decreases if diffuse functions and gradient corrections are employed. The value of y strongly depends on the level of calculation, and is about a factor of 2 larger if diffuse functions and gradient corrections are used. These values can be compared to those of Handy and co-workers.I0 The values of p at their best level (TZZP + diff)/BP and ours are in good agreement, and our value of a is about 5% larger than their value. Our components for a are in good agreement with their values.
Dixon and Matsuzawa
3972 The Journal of Physical Chemistry, Vol. 98, No. IS, 1994 TABLE 6
Calculated Geometry Parameters of the Conformers of Urea' Cz, CS(4 BPI BH/ BLYP/ BP/ MP-2/ BH/ BLYP/ DNP TZVP TZVP 6-31GISd DNP DNP DNP
r(C-0) 1.231 1.368 r(C-Nd 1.022 r(N I-H I ) ~ ( N I - H ~ ) 1.023 B(O-C-Nl) 121.9 116.2 B(N1-C-Nz) B(C-NI-HI) 124.7 116.2 B(C-Nl-Hz) ~ ( H I - N I - H ~ ) 119.1
1.241 1.388 1.021 1.020 122.4 115.2 124.4 117.0 118.6
1.234 1.383 1.013 1.012 122.4 115.1 124.5 116.6 118.9
1.228 1.374 1.007 1.007 122.8 114.4 124.2 116.9 118.9
1.238 1.405 1.026 1.025 122.4 114.9 117.6 111.5 112.5
1.229 1.379 1.027 1.026 122.2 115.3 118.6 111.8 114.0
1.233 1.391 1.016 1.016 122.4 115.0 120.6 113.9 115.9
C2b cs cs
CS Csb c 2 u
1.225 1.389 1.013 1.013 123.5 113.0 116.2 111.9 113.4
1.227 1.382 1.027 1.026 123.0 113.9 116.1 111.4 112.8
1.236 1.409 1.027 1.026 123.0 113.9 115.1 111.0 111.5
BP/ TZVP 1.231 1.395 1.017 1.017 123.0 114.0 117.7 113.0 113.9
expt2d
expt2b
1.270 1.326
1.243 1.351 0.998 0.995 121.5 117.0 118.1 119.8 122.1
121.0 118.1
BP/TZVP
MP-2/6-31G*5e
BH/DNP
CdC) BLYP/DNP
BP/TZVP
1.226 1.35 1 1.442 1.023 1.024 1.036 123.9 125.3 110.9 119.8 119.2 121.0 105.6 102.7
1.235 1.369 1.476 1.021 1.023 1.035 124.1 125.3 110.8 120.3 119.7 120.1 105.3 103.0
1.229 1.365 1.464 1.014 1.015 1.027 124.3 125.3 110.5 119.6 119.9 120.5 106.7 104.0
1.224 1.355 1.45 1
1.220 1.363 1.438 1.022 1.026 1.033 122.3 122.4 115.4 122.1 118.8 119.2 109.7 105.6
1.229 1.383 1.469 1.021 1.024 1.034 122.6 122.1 115.3 122.5 118.8 118.7 108.6 104.9
1.223 1.379 1.455 1.013 1.017 1.024 122.4 122.2 115.4 122.2 118.8 118.9 110.6 107.5
TABLE 7: Rotational Constants for Urea' A r(C-N) r(C=O) A(N-H)
c 2
BH/ DNP
BLYP/DNP
r(C-NI) r(C-N2) r(N I-H I ) ~(NI-H~) r(NrH3) B(O-C-N1) e(0-c-N~) B(Nl-C-N2) B(C-N 1-H I ) ~(C-NI-H~) WI-NI-H~) B( C-NI-H,) B(H~-N~-HS') 4a Units in angstroms and degrees for bond lengths and bond angles, respectively.
c 2
MP-2/ 6-31G*5d
BH/DNP r(C-0)
c 2
c 2
BLYP/ DNP
1.427 1.429 1.430
1.208 1.208 1.209
0.00 -0.01 -0.02
1.419 1.420 1.421
1.221 1.222 1.223
0.00 -0.01 4.02
1.416
1.220
0.00
11 228.7 11 232.6 11 232.0 11 176.6 11 231.3 11 231.4 11 231.4 11 265.6 11 242.9 11 242.8 11 242.7 11 283.5 11 233.3
+
B
C
10 375.8 10 364.9 10 367.2 10 818.4 10 369.2 10 371.0 10 372.7 10 751.3 10 363.1 10 365.0 10 367.0 10 824.5 10 369.4
5441.0 5438.0 5437.6 5547.4 5418.6 5418.6 5418.6 5529.4 5392.6 5393.0 5393.6 5524.6 5416.7
-0.01 1.221 1.417 C2" -0.02 1.222 1.418 C2" C2Db expt3 0 Units in angstroms for the bond lengths, and megahertz for the rotational constants. b Calculated with geometrical parameters at the BP/TZVP level as described in the text.
For @, it is clear that our value is somewhat larger in magnitude than their value and significantly larger than their MP-2 value. For urea, the value of the dipole moment depends on the geometry with the C2, structure having the largest value. The value of 1.69 au for the Cz, structure is in good agreement with the experimental value of 1.79 The lowest energy C2 conformer has a lower dipole moment of 1.43 au. The a b initio value for p is 2.0 au (DZV l s 2 ~ 4 d / 3 p / H F )and ~ ~ is 2.08 au at the INDO le~e1.3~ The value of a increases by 3.0 au (about 7%) if diffuse functions and gradient corrections are used. The value of @ is very sensitive to the level of calculation and to the conformation (the tensor components are given as supplementary material; see paragraph at end of paper). As found for formaldehyde, the value of @ decreases if diffuse functions and gradient corrections are included. This is true for all conformations. The largest value for @ is found for the Cbconformation, which is not surprising as the largest value of 1.1is also found here. The experimental @ has been reported to be 67-103 au in absolute values (EFISH, 1.064 pm),33 whereas our NLDFT absolute calculated values at the field strength of 0.010 au range from 95 to 119 au depending on the conformation, in good agreement with the range of experimental values. The ab initio SCF value
+
with the basis set of DZV ls2p4d/3p has been reported to be 37 a ~ , and 3 ~ thus, the S C F value is 1.6-2.4 times smaller than the experimental value. This is consistent with previous reports that the inclusion of electron correlation effects is required for calculating hyperpolarizabilities.22J4 For y, the NLDFT calculated values are again larger than the LDFT values by almost a factor of 2, and the value for the C,, conformer is larger than for the other conformers. The experimental y for urea is 72 000-120 000 au (EFISH, 1.064 pm),31 whereas the ab initio S C F value (DZV + ls2p4d/3p) is only 4700 au.32 The a b initio SCF value is much smaller than the DFT calculated values. Our NLDFT calculated values are -20 000 au, almost a factor of 3 smaller than the experimental value. On the basis of our previous results on substituted benzenes and polyene oligomers,8bthis difference is too large. One reason for this difference is that the measurement was made at 1,064 pm, whereas the calculated value is at zero frequency. Another possible reason is that the solvents used in the measurements are water, DMF, and DMSO, which can hydrogen bond with urea. This could lead to the large value of y that has been experimentally observed.35 Hyperpolarizabilities of Hz, Nz, CO, CH4, NH3, HzO, and HF. For completeness, the hyperpolarizabilities for Hz, N2, CO, CH4, NH3, HzO, and H F were calculated at the BH/DNP+2 and BLYP/DNP+2 levels (Table 12). Experimental9336 and the BH/DNP+2 values obtained by Guan et al. are also shown in Table 12. (The alignment of the molecules along the Cartesian axes is shown in Figure 2). We also compare these to the results who used a more extensive fitting procedure. On of Guan et the basis of our previous studies,* the hyperpolarizabilities were calculated with a finite field strength of 0.005 au. In addition to this field strength, calculations were also done with a strength of 0.020 au for comparison. For p, the dipole moment calculated by the Taylor series given in eq 2 as compared to the direct calculation from the density agree well with the largest difference found for CO which has the smallest dipole moment. As expected, the largest percentage difference between the BH and BLYP values is found for CO where the nonlocal corrections lower the dipole moment by about
Structures and Optical Properties of Urea
The Journal of Physical Chemistry, Vol. 98, No. 15, 1994 3973
TABLE 8: Calculated Geometry Parameters of NH3, Formamide, Formaldehyde, H2, Nz, CO, CHI, HzO, and HFa NH3 (C3v) r(N-H) e( H-N-H) NH3 (036) r(N-H) formamide (Cl(a)) r(C-0) ~(C-HI) r(C-N) r(N-H2) r(N-H3) O( 0-C-H I ) O(0-C-N) O(H 1-C-N) O(C-N-Ht) O(C-N-H3) O(H2-N-Hs) formamide (C,) r(C-0) r(C-Hi) r(C-N) ~(N-Hz) r(N-H3) O(0-C-H 1) O(0-C-N) O(C-N-HZ) O(C-N-H3) O (H2-N-H3) formamide (CI(b))c r(C-0) r(C-H I 1 r(C-N) 4N-W O(0-C-H 1) O(0-C-N) e( H 1-C-N) O(C-N-H*) O( H2-N-Hz’) formamide (Cl(c))c r(C-0) ~(C-HI) r(C-N) r(N-H2) O(0-C-H 1) O(0-C-N) O(H1-C-N) O(C-N-H2) O(H2-N-Hz’) formaldehyde r(C-0) r(C-H) O(HCH) H2 r(H-H) Nz r(N-N)
co
r(C-0) CH4 r(C-H) H2O r(0-H) e( H-0-H) HF r(H-F)
BH/DNP
BLYP/DNP
BPITZVP
ex&
1.034 104.4
1.033 104.5
1.023 106.9
1.012 106.7
1.017
1.014
1.006
1.223 1.120 1.355 1.026 1.029 123.0 124.4 112.5 119.7 116.4 118.2
1.233 1.114 1.378 1.024 1.025 123.1 124.6 112.3 118.6 116.3 116.5
1.227 1.115 1.368 1.015 1.017 123.2 124.8 112.1 121.4 119.1 119.4
1.224 1.120 1.352 1.025 1.028 122.7 124.7 121.3 118.9 119.8
1.235 1.113 1.371 1.022 1.024 123.2 124.7 121.5 119.3 119.1
1.212 1.118 1.431 1.039 120.5 125.8 113.6 106.4 102.1
1.222 1.1 11 1.467 1.037 121.1 126.0 113.0 106.2 102.4
1.208 1.125 1.427 1.039 120.2 122.9 116.9 107.4 102.0
1.218 1.119 1.459 1.038 121.0 122.5 116.5 106.5 102.8
1.213 1.124 116.7
1.225 1.120 117.2
1.208 1.116 116.5
0.769
0.751
0.741
1.113
1.118
1.094
1.145
1.152
1.128
1.103
1.100
1.094
0.985 103.5
0.986 103.5
0.957 104.5
0.956
0.957
0.917
TABLE 9 Calculated Vibrational Frequencies of H2, Nz, CO, CHd, H20, and HF in cm-l H2 N2
co CH4
HzO HF (I
1.219 1.098 1.352 1.002 1.002 112.8 120.0 118.5
Units in angstroms and degrees for bond lengths and bond angles, respectively. All experimental values from ref 27 except for CHzO which is from ref 28 and for formamide which is from ref Id. e Actual symmetry is C,. (I
60% (0.03 au). For the other compounds, the BLYP/DNP+2 values are almost the same as the BH/DNP+2 values with the BLYP/DNP+2 values smaller by 3 4 % ’ as compared to the BH/DNP+2 values, again a few hundredths of an au. The use of the DNP+2 basis set increases the magnitude of ~1as compared
BH/DNP
BLYP/DNP
expt‘
4166 2352 2133 1263(t) 1508(e) 2932(a) 3063(t) 1632(al) 3612(al) 3744(bl) 3805
4343 2312 2075 1327 1554 2936 304 1 1665 3545 3670 3748
440 1 2359 2170 1306 1534 2917 3019 1595 3657 3756 4138
Diatomic values from ref 29 and polyatomic values from ref 25.
to the DNP and DNP+ basis set values at the LDFT leve1;38.39 however, inclusion of gradient corrections at the DNP+2 level decreases p to give good agreement with the experimental values. The values obtained by Guan et al.9a in a more extensive fit to eq 2 at the local level differ from our values slightly with the largest percentage difference found for CO and also differ from those directly calculated from the density. For the polarizabilities (a,a,,, aYy,and azr),the calculated values with different field strengths do not differ significantly with respect to each other, although the values obtained with a field strength of 0.020 au are smaller by
