ARTICLE pubs.acs.org/JPCA
NMR Shielding Constants in PH3, Absolute Shielding Scale, and the Nuclear Magnetic Moment of 31P Perttu Lantto Department of Physics, University of Oulu, P.O. Box 3000, FI-90014 Oulu, Finland
Karol Jackowski, Wzodzimierz Makulski, and Mazgorzata Olejniczak Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
Michaz Jaszunski* Institute of Organic Chemistry, Polish Academy of Sciences, Kasprzaka 44, 01-224 Warsaw, Poland ABSTRACT: Ab initio values of the absolute shielding constants of phosphorus and hydrogen in PH3 were determined, and their accuracy is discussed. In particular, we analyzed the relativistic corrections to nuclear magnetic resonance (NMR) shielding constants, comparing the constants computed using the four-component DiracHartreeFock approach, the four-component density functional theory (DFT), and the BreitPauli perturbation theory (BPPT) with nonrelativistic HartreeFock or DFT reference functions. For the equilibrium geometry, we obtained σ(P) = 624.309 ppm and σ(H) = 29.761 ppm. Resonance frequencies of both nuclei were measured in gas-phase NMR experiments, and the results were extrapolated to zero density to provide the frequency ratio for an isolated PH3 molecule. This ratio, together with the computed shielding constants, was used to determine a new value of the nuclear magnetic dipole moment of 31P: μP = 1.1309246(50) μN.
1. INTRODUCTION Molecular properties, such as NMR shielding constants, can presently be calculated by a variety of ab initio methods. At the nonrelativistic level, shielding constants for small molecules can be computed with high accuracy. The convergence of the results with the treatment of electron correlation effects and with the extension of the basis set can be monitored, and thus error bars of the final results can be estimated. For comparison with experiment, one can add the zero-point vibrational and temperature corrections to the equilibrium values. It has been shown that, even for molecules such as PH3, the error bars of the nonrelativistic values are significantly smaller than the estimated relativistic corrections to the shielding constants.1 In that work, the relativistic effects in PH3 were determined considering the differences between DiracHartree Fock and HartreeFock results. However, such an approach is based on the assumption of additivity of electron correlation and relativistic effects, which obviously cannot be maintained for heavier nuclei. In the present work, we examine other methods enabling more accurate analysis of relativistic effects. In particular, we analyze in more detail the dependence of the computed relativistic corrections on the description of electron correlation and on the basis set. The new value of 31P shielding constant can be used to define the absolute shielding scale for phosphorus. In addition to the computed ab initio shielding constants, we present experimental gas-phase NMR results; we measured the resonance frequencies of both nuclei in phosphine as a function r 2011 American Chemical Society
of the gas density. To obtain the resonance frequencies for an isolated molecule (which is the system studied theoretically), these experimental results were next extrapolated to the zerodensity limit. Having obtained the shielding constants and resonance frequencies for both nuclei in this way, we next determined a new, accurate value of the nuclear magnetic dipole moment of 31P.
2. THEORETICAL BACKGROUND AND COMPUTATIONAL DETAILS 2.1. Methods. The nonrelativistic (NR) shielding constants were determined by applying coupled-cluster reference functions and analytic perturbation methods, adapted for the study of shielding constants. These methods were developed by Gauss and co-workers for the coupled-cluster singles and doubles (CCSD), as well as CCSD with a noniterative perturbative triples correction [CCSD(T)]2,3 and extended to higher coupled-cluster approximations by Gauss and Kallay.4,5 In particular, in these methods, gauge-including atomic orbitals (GIAOs)6 are applied in all nonrelativistic calculations. We concentrate in this work on the analysis of relativistic corrections to the shielding constants. The straightforward way Received: June 5, 2011 Revised: August 23, 2011 Published: August 23, 2011 10617
dx.doi.org/10.1021/jp2052739 | J. Phys. Chem. A 2011, 115, 10617–10623
The Journal of Physical Chemistry A to include the description of relativistic effects in calculations of the NMR shielding tensor is to utilize four-component relativistic theories based on the Dirac equation.7 In the present work, we chose the four-component DiracCoulomb (DC) Hamiltonian for that purpose and two methods, namely, DiracHartree Fock (DHF) and four-component density functional theory (DFT). The NMR shielding tensor was thus calculated from the four-component linear response equations in a perturbationdependent basis. In this formalism, this property is expressed by one term linear in both perturbations: that arising from the external magnetic field and that from the nuclear magnetic moments (contrary to nonrelativistic approaches, there is no expectation-value contribution; see eq 25 in ref 8). GIAOs were systematically used in these calculations, and in addition, a new scheme that simulates the explicit magnetic balance9 was used to create the small component basis set. As the DFT method takes into account some electron correlation effects, the difference between the results obtained with the four-component DFT and four-component HF approaches provides a prediction of the magnitude of electron correlation effects within the four-component relativistic theory. At the same time, by performing calculations with four-component LevyLeblond (LL) Hamiltonian,10 which can be described briefly as the nonrelativistic limit of DiracCoulomb Hamiltonian, the magnitude of relativistic effects can be estimated for both DFT and HF methods. Relativistic effects can then be determined as the difference between the four-component DC results and the corresponding four-component LL results (with the latter practically equal to the nonrelativistic results). The leading-order BreitPauli perturbation theory (BPPT)1113 provides an approach complementary to four-component DC HF/DFT methods to conduct an analysis of relativistic corrections to the shielding constants. BPPT treats relativistic phenomena on an equal footing with magnetic operators as perturbations acting on a spin-free NR reference state, for which the basis set limit is easier to reach, and moreover, both DFT and post-HF ab initio correlated wave function methods already exist. For the isotropic shielding constant, there are 14 leading-order oneelectron BPPT terms, including both scalar relativistic (SR) and spinorbit (SO) interactions, which can be interpreted in terms of familiar NR concepts. (A more detailed description of these terms and contributing operators can be found in the original articles11,12 and a review.13) Previous studies of shielding tensors of heavy14,15 and light16 atoms have shown that the BPPT contributions can be divided into “core” and “shift” contributions. The former are practically isotropic and insensitive to electron correlation and ligand effects.17 Therefore, the relativistic effect in the chemical shift and anisotropic parameters is mainly due to shift contributions: second-order kinetic energy (p-KE/OZ), third-order singlet mass-velocity (p/mv) and Darwin (p/Dar) corrections to paramagnetic shielding, and third-order triplet SO corrections due to Fermi contact (FC-I) and spin-dipole (SD-I) hyperfine interactions. (For the latter, we also include the two-electron counterparts known to be important in very precise studies of light-atom shieldings.16) All of these shift terms are also responsible for the possible cross-coupling of relativity and electron correlation. In particular, the SO-I contributions depend on excitations to the triplet excited states that are poorly described at the uncorrelated HF level and can thus require correlation treatment to provide reliable results.1416
ARTICLE
Table 1. σ(31P) Shielding Constant in PH3 (ppm)a u-VDZb u-wCV5Zc
co-bd
co-re
HF/co-rf
Four-Component Theory DiracCoulomb (DC) 629.399
614.676
616.395 615.691 603.078
LevyLeblond (LL)
614.346
597.902
597.974 597.205 584.105
15.053
16.774
DC LLg
18.421
18.486
18.973
BreitPauli Perturbation Theory Core Contributions con
12.817 13.130 13.166 13.165 12.957
d-KE
10.232 10.392 10.409 10.409 10.242
p-OZ p/OZ-KE d/mv
0.370
0.372
0.372 0.372 0.364
1.636 17.306
1.653 17.725
1.654 17.784
1.655 17.800
1.641 17.516
d/Dar
9.786 10.110 10.144 10.152 10.050
FC-II(1)
7.167
7.408
SD-II(1)
0.126
0.262
1.362
1.351
1.330
FC/SZ-KE
41.109
42.475
42.612
42.657
42.078
p-KE/OZ
2.701
3.039
3.341
3.301
3.300
3.945 0.602
4.762 0.253
7.432 7.442 7.392
Shift Contributions p/mv p/Dar
5.157 5.122 5.122 0.202 0.224 0.224
FC-I(1)
0.317
0.392
FC-I(2)
0.481
0.502
SD-I(1)
0.739
0.877
0.966 0.974 0.965
SD-I(2)
0.004
0.007
0.014 0.013 0.018
∑(BPPT)h NR TOTAL
17.058
17.733
614.349 631.406
597.909 615.642
0.388
0.378
0.238
0.499 0.499 0.457
18.780
18.771
18.310
597.972 597.207 584.106 616.752 615.978 602.416
a DFT calculations with KT2 functional if not otherwise noted. b Uncontracted cc-pVDZ basis set; 62 basis functions. c Uncontracted ccpwCV5Z basis set; 370 basis functions. d Best complete optimized basis set for both P (26s22p18d8f) and H (19s7p4d), providing basis set limit result for NR and each BPPT contribution; 418 basis functions. e Reduced co-r basis for P (22s18p14d6f) and H (11s3p3d); 293 basis functions. f HartreeFock calculation with co-r basis. g Relativistic effect by four-component theory. h Relativistic effect by BPPT theory (sum of all terms, including both one- and two-electron third-order spinorbit FC-I and SD-I contributions).
The shift terms are also the main source of the relativistic effects due to the nearby heavy atom on the light-atom shielding (HALA effect),16 whereas the relativistic heavy-atom effects on the shielding of the heavy atom itself (HAHA effect) include large contributions from the second-order magnetic-fielddependent SO [SO-II(1) = FC-II(1) + SD-II(1)] terms, as well as the cross-term of the Fermi contact hyperfine interaction with the relativistically modified spin-Zeeman interaction (FC/SZ-KE).17 DFT calculations of phosphorus chemical shifts indicate that the performance of different functionals varies significantly with the basis set (see, e.g., refs 18 and 19). In both relativistic approximations at the DFT level, we chose the KealTozer KT2 functional, which has been shown to provide successful descriptions of the correlation effects in nonrelativistic calculations of shielding constants.20 This choice was confirmed by the present results for σ(P): Using the KT2 functional, we recovered 10618
dx.doi.org/10.1021/jp2052739 |J. Phys. Chem. A 2011, 115, 10617–10623
The Journal of Physical Chemistry A more than 50% of the correlation effects determined by the CCSD(T) approach (see the nonrelativistic results in section 3.1 and Table 1 below). We also determined the relativistic corrections to σ(31P) by applying a few other functionals, although for PH3 they do not provide reliable estimates of the correlation effects for NR shielding with currently used practically converged basis sets. In this work, we used a sequence of computer programs. The nonrelativistic coupled-cluster calculations were performed using the CFOUR package.21 Next, we estimated the corrections to the CCSD(T) approach by applying the MRCC22 program. To determine the relativistic effects, an experimental version of the DIRAC package23 including four-component DHF linear response with GIAOs8 and newly implemented four-component DFT linear response with GIAOs24 was used, and a local (Oulu) version of the Dalton code was applied in all BPPT calculations.25 2.2. Basis Sets. We first used Dunning’s basis sets;26 we report herein the results obtained with uncontracted versions of ccpVDZ and cc-pwCV5Z, our largest set of this type (the notation u-VDZ and u-wCV5Z is used in what follows). In addition, we used basis sets optimized according to the completeness-optimized (co) basis-set paradigm.27 Our “best” co-b basis sets for third- and first-row nuclei (P/H: 26s22p18d8f/19s7p4d) were generated to reach the Gaussian basis set limit for nonrelativistic NMR shielding and spinspin coupling tensors, as well as relativistic BPPT corrections to shielding constants.28,29 This was carried out (as also done for the second-row element C30) by maximizing the overlap with a test Gaussian primitive, with its exponent sweeping a wide enough range for convergence of each parameter. The exponent ranges, [ζmin, ζmax] used in the Kruununhaka code31 to generate the co basis sets for P28 are: s-type functions [0.01, 1.0 108], p-type functions [0.01, 1.0 105], d-type functions [0.1, 31622], and f-type functions[0.316, 31.6], and those for H29 are: s-type functions [0.0316, 1.0 105], p-type functions [0.1, 10.0], and d-type functions [0.1, 3.16]. The “reduced” co-r basis sets (P/H: 22s18p14d6f/11s3p3d) were obtained by reducing the number of basis functions in the exponent ranges of the co-b set while keeping the resulting errors of properties very small. These co basis sets are not characteristic for specific atoms; instead, they are universal provided that the exponent range is sufficient for given properties. Hence, they are expected to provide similar quality of the studied parameters for different elements in the same row of the periodic table.32 We used the same basis sets in both four-component and BPPT calculations, but in the former, we systematically used GIAOs, whereas in the latter, GIAOs were used only in the NR part whereas the relativistic effects were computed with fixed common gauge origin at phosphorus. However, this is the natural gauge origin for PH3, and for large enough basis sets, this difference does not play a major role (see below).
3. RESULTS AND DISCUSSION 3.1. Nonrelativistic Shielding Constants. The dependence of the shielding constants in PH3 on the description of electron correlation effects and on the basis set was examined in refs 1 and 33. The best nonrelativistic values determined in ref 1 for the equilibrium geometry at the CCSD(T)/cc-pCVQZ level were σ(P) = 605.831 ppm and σ(H) = 29.514 ppm; the corresponding values at the CCSD(T)/cc-pwCVQZ level (with 1s of P not correlated) are σ(P) = 605.002 ppm and σ(H) = 29.558 ppm.33
ARTICLE
Table 2. σ(1H) Shielding Constant in PH3 (ppm)a u-VDZ
u-wCV5Z
co-b
co-r
HF/co-r
Four-Component Theory DiracCoulomb
29.373
29.357
29.697
29.691
29.844
LevyLeblond
30.036
29.591
29.589
29.573
29.659
DC LL
0.663
0.234
0.108
0.119
0.185
BreitPauli Perturbation Theory Core Contributions con
0.002
0.003
0.003
0.003
0.003
d-KE
0.001
0.001
0.001
0.001
0.001
p-OZ
0.000
0.001
0.001
0.001
0.001
p/OZ-KE d/mv
0.001 0.049
0.001 0.052
0.001 0.052
0.001 0.052
0.001 0.041
d/Dar
0.046
0.048
0.048
0.048
0.042
FC-II(1)
0.003
0.003
0.003
0.003
0.004
SD-II(1)
0.000
0.000
0.000
0.000
0.000
FC/SZ-KE
0.002
0.002
0.002
0.002
0.004
p-KE/OZ
0.001
0.001
0.001
0.001
0.001
p/mv p/Dar
0.045 0.037
0.054 0.044
0.054 0.044
0.054 0.044
0.054 0.044
Shift Contributions
FC-I(1)
0.222
0.284
0.295
0.294
0.280
FC-I(2)
0.052
0.068
0.070
0.070
0.068
SD-I(1)
0.004
0.010
0.010
0.010
0.009
SD-I(2)
0.001
0.002
0.002
0.002
0.002
∑(BPPT) NR total a
0.164
0.213
0.221
0.221
0.203
30.036 30.200
29.591 29.804
29.589 29.810
29.573 29.793
29.659 29.862
See foonotes to Table 1.
We have now obtained new nonrelativistic values, computed at a geometry optimized at the CCSD(T)/aug-pVQZ level, and applying the CCSD(T) approach with the cc-pwCV5Z basis set for the shielding constants (and in all of the calculations in this work, all of the electrons are correlated). Our new values are σ(P) = 606.110 and σ(H) = 29.547 ppm. These values do not differ significantly from the previous results, differing by ∼1.0 ppm or less from the previous results for phosphorus and by less than 0.05 ppm for the H nucleus, and it appears that they would not be changed significantly with a better basis set. In addition, we have now estimated—using small basis sets— the roles of triple and quadruple excitations in the cluster operator. Using the cc-pwCVTZ basis set, we found the differences between CCSDT and CCSD(T) values to be 0.887 ppm for phosphorus and 0.005 ppm for hydrogen shielding. The effect of quadruple excitations was estimated using the cc-pVDZ basis set as 0.306 ppm for phosphorus and 0.002 ppm for hydrogen shielding. With this basis set, for the CCSDT CCSD(T) difference, we obtained 0.671 ppm, in fair agreement with the cc-pwCVTZ result. Including these minor corrections, we found that the total electron correlation contribution to the shielding constant is +20.494 ppm for σ(P) and 0.221 ppm for σ(H). 3.2. Relativistic Effects. For 31P shielding in PH3, the relativistic correction was estimated in ref 1 to be 15.36 ppm. 10619
dx.doi.org/10.1021/jp2052739 |J. Phys. Chem. A 2011, 115, 10617–10623
The Journal of Physical Chemistry A However, this value was derived as the difference between DiracHartreeFock (DHF) and HartreeFock results; thus, the calculation of the total shielding constant was based on the assumption that electron correlation and relativistic effects are additive. Moreover, the DHF and HF results were obtained for the common gauge origin at the P atom; it was not possible at that time to use GIAOs in four-component calculations, and the convergence of the results with the extension of the basis set was very slow. The computed relativistic corrections to the shielding constants are reported in Tables 1 and 2. We present the KT2 results, because the difference between DFT and HF LevyLeblond phosphorus shielding constants for this functional, +13.1 ppm for the co-r basis set, is closest to our best estimate of the electron correlation effect within the nonrelativistic formalism. The first four columns of each table illustrate the basis set dependence of the relativistic corrections at the correlated (DFT) level—computed as the difference between DC/DFT and LevyLeblond/DFT results or directly in the BPPT approach. For comparison, in the last column, we present the relativistic corrections calculated at the uncorrelated (HF) level, equal to DC/DHF LevyLeblond/HF, or computed by applying the HF reference state within the BPPT approach. One can see in Table 1 that, for the phosphorus shielding constant, both Dunning-type basis sets give qualitatively correct estimates of the relativistic correction, although the larger uncontracted cc-pwCV5Z result is still about 1.5 ppm below the value obtained with the optimized co-b basis sets. This underestimation is slightly smaller at the BPPT level of theory, illustrating its easier basis set convergence. The ∼1 ppm error in the BPPT relativistic effect on σ(P) is mostly due to the underestimated SD-II(1) contribution, which requires tight d-type polarization functions. Also, to fully converge the p-KE/OZ and p/mv corrections in the co basis, p-type basis functions tighter than present in the u-wCV5Z basis set were needed. For co-r and co-b basis sets the four-component calculations and BPPT approach give similar results. Somewhat more stringent demands for the basis set quality at the four-component level can be seen for the proton shielding in Table 2: Only the co basis sets lead to a positive relativistic correction to σ(H), in agreement with the BPPT results. The more economical reduced co-r basis set outperforms the u-wCV5Z set, providing nonrelativistic σ(P) and σ(H) shielding constants close to the basis set limit, as well as proper descriptions of the relativistic effects at both the four-component and BPPT levels of theory. We note finally that, although the total relativistic corrections to the two shielding constants are almost reproduced even with the modest uncontracted cc-pVDZ set, an analysis of the individual BPPT terms shows that this is due to mutual cancellation of significant errors. Most of the relativistic effects on absolute σ(P) shielding constant are due to the HAHA effects, that is, P's own core contributions, equal to 21.890 ppm in the co-b basis set. We note that the core terms are large and that all of them have to be calculated (although there is a significant cancellation of these contributions); one can also see the particular importance of the above-mentioned FC/SZ-KE contribution.17 Typically, the core terms are not sensitive to electron correlation, and thus, the HF results are very close to the DFT ones. Additionally, the core terms are not affected much by the chemical environment (hence, the chemical shift due to ligands is almost solely mediated by the shift terms, which are inherently zero for a spherically symmetric
ARTICLE
atom or ion14,17). These shift terms decrease the relativistic effect to P shielding constant by 3.109 ppm, that is, about 14%. The significant two-electron FC-I(2) term that overcompensates the corresponding one-electron FC-I(1) contribution is also noteworthy. We observe that the partition into core and shift contributions is practically independent of the basis set. The relativistic correction to σ(H) is mostly due to the HALA effect of phosphorus; the core contribution is practically negligible for hydrogen. The SO-I shift contributions, especially FCI(1), are mainly responsible for the relativistic effects. Our HF/co-r values for the phosphorus shielding constants in Table 1 are fairly close to 597.03 ppm, the result obtained by Maldonado and Aucar34 within the fully relativistic polarization propagator formalism at the random phase approximation (RPA) level (with the UKB prescription and the largest 15s12p11d6f basis set). We note that the triplet instability effects in PH3 appear to be rather small, as HF theory gives a sensible approximation of the spinorbit contributions and, consequently, a reasonable estimate of the total relativistic effects for both P and H shielding constants. We confirmed this by computing the relativistic corrections with a series of other DFT functionals (BLYP, B3LYP, BHandHLYP, PBE, and PBE0; see refs 3540), using BPPT and/or DC LL approximations and the co-r basis set. The differences between various results are practically negligible: we found that all of the results for σ(P) are in the range 18.0518.58 ppm and all of those for σ(H) are in the range 0.120.19 ppm. For some of these functionals, the nonrelativistic phosphorus shielding constants are—in contrast to KT2 value—significantly worse than the HF result (see also ref 18). Nevertheless, all of the computed relativistic corrections to the shielding constants are very similar, which shows that, for the shielding constants in PH3, the correlation and relativistic effects are practically additive. This indicates, moreover, that similar values of the relativistic corrections should be obtained using correlated wave functions as the reference. We also verified by moving the common gauge origin from P to the shielded proton that the possible errors due to gauge origin dependence are negligible. Finally, we discussed above only the tensor averages of the relevant contributions to the shielding constants; for some of the contributions, the signs of all of the individual tensor components differ [e.g., for the FC-I(1) versus FC-I(2) contributions to σ(P)], and thus, the total FC-I becomes even smaller, whereas the SD-I term is the main SO-I contribution.
4. NUCLEAR MAGNETIC DIPOLE MOMENT OF 31P Magnetic dipole moments of many nuclei have been established from NMR spectra.41 Comparing the experimental resonance frequencies for two different spin-1/2 nuclei X and Y in the same external magnetic field, νX and νY, one can determine the magnetic moment μY from the equation μY ¼
νY ð1 σX Þ μ νX ð1 σ Y Þ X
ð1Þ
assuming that μX and the values of the absolute shielding constants of both nuclei are known. We stress that, to apply eq 1, one needs absolute shielding constants; chemical shifts are useless in this context. One needs also an accurate value of μX, and the preferred reference nucleus is hydrogen, because the proton magnetic moment μH = 2.792847356(23) μN42 is known with high accuracy (and in PH3 phosphorus and the protons are 10620
dx.doi.org/10.1021/jp2052739 |J. Phys. Chem. A 2011, 115, 10617–10623
The Journal of Physical Chemistry A
ARTICLE
Table 3. Total Shielding Constants (ppm) σ(31P)
σ(1H)
equilibrium geometry nonrelativistic values, CCSD(T)a
606.110
29.547
nonrelativistic values, CCSDT-CCSD(T)
0.887
0.005
0.306
0.002
18.780
0.221
total, equilibrium geometry
624.309
29.761
ZPV1 c
9.227
0.436
300 K1
0.324
0.020
total, 300 K
614.758
nonrelativistic values, CCSDTQ-CCSDT relativistic correction, BPPTb
29.305d
a
At the HF level, the corresponding values are 585.035 and 29.780 ppm. At the KT2 level, in the four-component approach, the corresponding corrections are18.421 and 0.108 ppm. c The corresponding values in ref 33 are 8.820 and 0.432 ppm. d The experimental value of σ300 K(H) is 29.237 ppm. b
in the same molecule). We also treat 3He as the reference nucleus, using μHe = 2.127625306(25) μN (derived from the moment of shielded 3He42 and an accurate helium atom shielding constant, 59.96743 ppm43). The shielding constants σX and σY were taken from the discussed ab initio calculations, and the resonance frequency ratio was obtained from the described below new spectra of gaseous PH3, which enabled us to determine this ratio for an isolated PH3 molecule. The generally used absolute shielding scale for 31P was obtained long ago by considering the NMR data for the same molecule. This scale was defined by adding to the calculated diamagnetic term the paramagnetic term derived from the measured spin-rotation constant in PH3.44,45 The diamagnetic term, 981.01 ppm, was estimated at the HartreeFock level and— together with the paramagnetic contribution extracted from spinrotation constant—led to 599.93 ppm as the best equilibrium value. It appears that ab initio values are currently more accurate, because neither the calculated diamagnetic term nor the standard equation relating the spin-rotation constant to the paramagnetic shielding take into account any relativistic effects. We verified that the nonrelativistic diamagnetic term was rather accurately estimated, as a corresponding CCSD(T)/cc-pwCV5Z calculation (with the gauge origin fixed at phosphorus) gave 981.65 ppm. A summary of the computed shielding constants is given in Table 3. In addition to the nonrelativistic contributions, we include the relativistic effects and—for comparison with experiment—the zero-point vibrational (ZPV) and temperature effects. For hydrogen, the absolute shielding scale is known; hence, the computed shielding constant can be compared with an experimental value. For phosphorus, the present results can be used to define an improved absolute shielding scale. In addition to the determination of absolute shielding constants, eq 1 requires the measurement of 1H and 31P resonance frequencies in a PH3 molecule free from intermolecular interactions. Modern NMR spectrometers allow the reading of absolute frequency with high accuracy, as the external magnetic field remains unchanged due to the stable superconducting magnet and deuterium lock system (thus, no analysis of the accuracy of the frequency ratio in eq 1 is needed). The appropriate experiment is performed in the gas phase where the intermolecular effects are weak and can be analyzed by observing the density
dependence of the resonance frequency. As shown in numerous earlier studies, this dependence is usually linear, and by extrapolating the experimental points to the zero-density limit, one can determine the resonance frequency for an isolated molecule.46 In this study, we examined the 1H and 31P frequencies in gaseous phosphine within the same experimental framework. The glass samples additionally contained small amounts of 3He (99.9995%, Aldrich) from lecture bottles were used to prepare gas samples without further purification. The resonance frequencies were measured on a Varian INOVA 500 spectrometer at 300 K. 1H and 31P spectra were acquired with the standard two-channel Varian switchable 5 mm probe, whereas 3 He spectra were obtained in a self-reconstructed special helium probe.48 The extrapolation of the experimental data to zero density gives νP = 202.5950278, νH = 500.6067303, and νHe = 381.3566058 MHz. Substituting these values into eq 1 and treating 1H as the reference nucleus, we find that the magnetic dipole moment of 31P is 1.13092456 μN. We can additionally verify this result using the 3He data as input, and we obtain 1.13092468 μN. When both magnetic moments are known, one can invert eq 1 and compute the shielding of a nucleus of interest from the other data. To check the consistency of our results, we applied the μP value determined with 1H as the reference to recompute the shielding of the helium atom, and we obtained 59.904 ppm—differing by ∼0.06 ppm from the correct value. Applying the standard literature value of μP = 1.13160(3) μN41 in the same way, we obtain σ(He) = 53.7 ppm, for which the error is approximately 114 ppm. We also estimated the error bars of the new 31P magnetic moment. Assuming that the error in the shielding of phosphorus does not exceed 4 ppm and that that in the shielding of hydrogen is 0.2 ppm, we find μP = 1.1309246(50) μN; the estimate of the error bars obtained from the 3He data is similar.
5. CONCLUSIONS We provide new estimates of the total shielding constants in phosphine. The nonrelativistic values at the equilibrium geometry, computed at a somewhat more advanced level than previously, are similar to the earlier results. Our primary aim in this work was to evaluate the relativistic corrections and to analyze the accuracy of the results. We observed very satisfying agreement between different estimates of the relativistic effects. For phosphorus, using the KT2 functional, we obtained 18.421 ppm with the four-component approach and 18.780 ppm with the BPPT. It appears therefore that the error in this value does not contribute significantly to the estimated error bars of the total shielding constant. 10621
dx.doi.org/10.1021/jp2052739 |J. Phys. Chem. A 2011, 115, 10617–10623
The Journal of Physical Chemistry A Adding these relativistic corrections to the coupled-cluster nonrelativistic results, we obtained 624.3 ppm for the total shielding constant of phosphorus at the equilibrium geometry. Both approximations that we used—the four-component DC approach and the BPPT—indicated that the electron correlation effects and relativistic effects in PH3 are practically additive; thus, one should not expect a significant change of relativistic correction with a more advanced correlated reference wave function. A detailed analysis of the BPPT terms revealed that this additivity arises from two facts: first, the main part of the relativistic correction is caused by the core contributions, insensitive to the electron correlation, and second, the correlation-dependent shift contributions are much smaller and, in the case of PH3, only slightly affected by electron correlation. Including the temperature effects, for the total shielding constant of phosphorus at 300 K, which can be used to analyze the experimental data, we found σ(P) = 614.758 ppm. The best calculated value in ref 1 was 611.640 ppm. However, the best estimate, obtained considering the convergence of different computed contibutions to the total value (in particular, the poorly converging with the basis set relativistic contribution), was 614.2 ppm. This estimate is in good agreement with the present result, which includes a much more accurate relativistic contribution to the shielding constant. There is a discrepancy of more than 24 ppm between the generally used value and the calculated new ab initio phosphorus absolute shielding constant. We attribute this discrepancy to the derivation of the literature value as a sum of the nonrelativistic diamagnetic term and the paramagnetic term deduced from the spin-rotation constant; it does not appear likely that any of the approximations used in the present ab initio calculation is responsible for this difference. An estimate of the shielding constant in gaseous phosphine serves to establish the absolute shielding scale. The generally accepted standard reference for phosphorus NMR data is the shielding in 85% water solution of H3PO4. Applying our result, 614.8 ppm at 300 K, and the value of the chemical shift that we measured, 263.24 ppm (it was 266.10 ppm in ref 45), we obtained 351.6 ppm for the shielding of phosphorus in H3PO4. This value can be used to analyze the commonly measured chemical shifts and the computed shielding constants in other molecules. On the other hand, absolute shielding constants were needed to analyze the nuclear magnetic moment. We measured the required resonance frequency ratio for two nuclei in the same external field and for precisely the same system for which the shielding constants are determined. Considering the accuracy of the calculations and of the experiment, we believe that the new value of the phosphorus magnetic dipole moment is more accurate than the available literature result.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT P.L. is an Academy Researcher of the Academy of Finland and with the Finnish Centre of Excellence in Computational Molecular Science. The computational resources were partially provided by CSC-IT Center for Science (Espoo, Finland) and
ARTICLE
partially by the Norwegian Supercomputing Program (Notur). M.O. was supported by the Foundation for Polish Science (project operated within the Foundation for Polish Science MPD Programme cofinanced by the EU European Regional Development Fund) and by Polish Grant N N204 116539. We acknowledge support by the Polish Ministry of Science and Higher Education Research Grant N N204 244134 (20082011).
’ REFERENCES (1) Antusek, A.; Jaszu nski, M. Mol. Phys. 2006, 104, 1463. (2) Gauss, J.; Stanton, J. F. J. Chem. Phys. 1995, 102, 251. (3) Gauss, J.; Stanton, J. F. J. Chem. Phys. 1996, 104, 2574. (4) Gauss, J. J. Chem. Phys. 2002, 116, 4773. (5) Kallay, M.; Gauss, J. J. Chem. Phys. 2004, 120, 6841. (6) London, F. J. Phys. Radium 1937, 8, 397. (7) Saue, T. Post DiracHartreeFock Methods—Properties. In Relativistic Electronic Structure Theory. Part 1. Fundamentals; Schwerdtfeger, P., Ed.; Elsevier: Amsterdam, 2002; Chapter 7, p 332. (8) Ilias, M.; Saue, T.; Enevoldsen, T.; Jensen, H. J. Aa. J. Chem. Phys. 2009, 131, 124119. (9) Komorovsky, S.; Repisky, M.; Malkina, O. L.; Malkin, V. G.; Malkin Ondik, I.; Kaupp, M. J. Chem. Phys. 2008, 128, 104101. (10) Visscher, L.; Saue, T. J. Chem. Phys. 2000, 113, 3996. (11) Manninen, P.; Lantto, P.; Vaara, J.; Ruud, K. J. Chem. Phys. 2003, 119, 2623. (12) Manninen, P.; Ruud, K.; Lantto, P.; Vaara, J. J. Chem. Phys. 2005, 122, 114107. erratum. J. Chem. Phys. 2006, 124, 149901. (13) Vaara, J.; Manninen, P.; Lantto, P. Perturbational and ECP Calculation of Relativistic Effects in NMR Shielding and SpinSpin Coupling. In Calculation of NMR and EPR Parameters. Theory and Applications; Kaupp, M., B€uhl, M., Malkin, V. G., Eds.; Wiley-VCH: Weinheim, Germany, 2004; Chapter 13, p 209. (14) Lantto, P.; Vaara, J. J. Chem. Phys. 2007, 127, 084312. (15) Straka, M.; Lantto, P.; R€as€anen, M.; Vaara, J. J. Chem. Phys. 2007, 127, 234314. (16) Kantola, A. M.; Lantto, P.; Vaara, J.; Jokisaari, J. Phys. Chem. Chem. Phys. 2010, 12, 2679. (17) Lantto, P.; Romero, R. H.; Gomez, S. S.; Aucar, G. A.; Vaara, J. J. Chem. Phys. 2006, 125, 184113. (18) van W€ullen, C. Phys. Chem. Chem. Phys. 2000, 2, 2137. (19) Chernyshev, K. A.; Krivdin, L. B. Russ. J. Org. Chem. 2010, 46, 785. (20) Keal, T. W.; Tozer, D. J. J. Chem. Phys. 2003, 119, 3015. (21) CFOUR, a quantum chemical program package written by Stanton, J. F.; Gauss, J.; Harding, M. E.; Szalay, P. G. with contributions from Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Christiansen, O.; Heckert, M.; Heun, O.; Huber, C.; Jonsson, D.; Juselius J.; Klein, K.; Lauderdale, W. J.; Matthews, D.; Metzroth, T.; O’Neill, D. P.; Price, D. R.; Prochnow, E.; Ruud, K.; Schiffmann, F.; Stopkowicz, S.; Tajti, A.; Varner, M. E.; Vazquez J.; Wang, F.; Watts, J. D. and the integral packages MOLECULE (J. Alml€ of and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van W€ullen. For the current version, see http://www.cfour.de. (22) MRCC, a string-based quantum chemical program suite written by M. Kallay. See also Kallay, M.; Surjan, P. R. J. Chem. Phys. 2001, 115, 2945, as well as http://www.mrcc.hu. (23) DIRAC, a relativistic ab initio electronic structure program, Release DIRAC10 (2010), written by Saue, T.; Visscher, L.; Jensen, H. J. Aa.; with contributions from Bast, R.; Dyall, K. G.; Ekstr€om, U.; Eliav, E.; Enevoldsen, T.; Fleig, T.; Gomes, A. S. P.; Henriksson, J.; Ilias, M.; Jacob, Ch. R.; Knecht, S.; Nataraj, H. S.; Norman, P.; Olsen, J.; Pernpointner, M.; Ruud, K.; Schimmelpfennig, B.; Sikkema, J.; Thorvaldsen, A.; Thyssen, J.; Villaume, S.; Yamamoto, S. (see http:// dirac.chem.vu.nl). (24) Olejniczak, M.; Bast, R.; Saue, T.; Pecul, M. Unpublished (2011). 10622
dx.doi.org/10.1021/jp2052739 |J. Phys. Chem. A 2011, 115, 10617–10623
The Journal of Physical Chemistry A
ARTICLE
(25) DALTON, a molecular electronic structure program, release 2.0 (2005). See http://www.kjemi.uio.no/software/dalton/dalton.html. (26) Peterson, K. A.; Dunning, T. H., Jr. J. Chem. Phys. 2002, 117, 10548. (27) Manninen, P.; Vaara, J. J. Comput. Chem. 2006, 27, 434. (28) Lantto, P. University of Oulu, Oulu, Finland. Unpublished (2011). (29) V€ah€akangas, J.; Lantto, P.; Vaara, J. University of Oulu, Oulu, Finland. Unpublished (2011). (30) Ik€al€ainen, S.; Lantto, P.; Manninen, P.; Vaara, J. Phys. Chem. Chem. Phys. 2009, 11, 11404. (31) Kruununhaka Basis Set Tool Kit, Release 2.0 (2011), written by Manninen, P.; Lehtola, J. See http://www.chem.helsinki.fi/∼manninen/ kruununhaka. (32) Ik€al€ainen, S.; Romalis, M. V.; Lantto, P.; Vaara, J. Phys. Rev. Lett. 2010, 105, 153001. (33) Puzzarini, C. Theor. Chem. Acc. 2008, 121, 1. (34) Maldonado, A. F.; Aucar, G. A. Phys. Chem. Chem. Phys. 2009, 11, 5615. (35) Becke, A. D. Phys. Rev. A 1988, 38, 3098. (36) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785. (37) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (38) Becke, A. D. J. Chem. Phys. 1993, 98, 1372. (39) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (40) Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158. (41) Raghavan, P. At. Data Nucl. Data Tables 1989, 42, 189. (42) Mohr, P. J.; Taylor, B. N.; Newell, D. B. Rev. Mod. Phys. 2008, 80, 633. (43) Rudzinski, A.; Puchalski, M.; Pachucki, K. J. Chem. Phys. 2009, 130, 244102. (44) Jameson, C. J.; de Dios, A. C.; Jameson, A. K. J. Chem. Phys. 1991, 95, 9042. (45) Jameson, C. J.; de Dios, A.; Jameson, A. K. Chem. Phys. Lett. 1990, 167, 575. (46) Jackowski, K.; Jaszunski, M. Conc. Magn. Reson. Part A 2007, 30, 246. (47) Jackowski, K.; Jaszunski, M.; Wilczek, M. J. Phys. Chem. A 2010, 114, 2471. (48) Jackowski, K.; Jaszunski, M.; Kamienski, B.; Wilczek, M. J. Magn. Reson. 2008, 193, 147.
10623
dx.doi.org/10.1021/jp2052739 |J. Phys. Chem. A 2011, 115, 10617–10623
