- Email: [email protected]
The 13C nuclear magnetic shielding constants at the singlet excited states in CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO
Chemical Physics Letters 724 (2019) 86–89
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
The 13C nuclear magnetic shielding constants at the singlet excited states in CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO
T
Michiko Atsumi Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Norway
H I GH L IG H T S
state NMR shielding constants are computed at the correlated CASSCF level. • Excited excited state shielding constants are deshielded compared to ground state. • Singlet • Large paramagnetic effects are likely due to population of 2p orbitals.
A B S T R A C T
Quantum-chemical calculations of excited state nuclear shielding constants are reported for singlet states in CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO. The multiconfigurational self-consistent field method is used to account for static correlation in the excited states. With one exception, the molecules considered had larger shielding constants in their excited states than in their ground states. This tendency towards deshielding is rationalized using the sum-over-states expression for magnetic response.
1. Introduction Nuclear magnetic resonance (NMR) is presently used to measure the structure of molecules in their electronic ground state. NMR can probe the events on a micro- to nano-second time scale. However, the lifetime of the singlet excited state lies in the picosecond and subpicosecond range. This makes it difficult to measure the molecular structure in the excited states since NMR measurements and the excited state phenomena have different time scales. In this report, we will investigate the NMR spectroscopy of the excited states using quantum chemical methods. Computational techniques will be used to model the electronic wave functions and magnetic properties of ground states and excited states. The differences in structure between ground state and excited state manifest themselves in different values of the nuclear shielding constants. This quantity is sensitive to molecular and electronic structure, and the chemical shift of NMR resonance indicates a change in the electronic structure at the associated nuclei. One advantage of quantum-chemical calculations is that they can address systems and properties that are difficult to access experimentally. The NMR technique utilizes that in the presence of an external magnetic field a nuclear spin state will be aligned with the field. For example, for doublet states (as found in the 1H nuclei) the spin is either parallel or anti-parallel with respect to the external field. The energy difference of two such states, which is normally in the radio-frequency
range, is probed in NMR experiments. However this is not probed in a direct way. Rather, the nuclear spins are exposed to a strong constant magnetic field and allowed to come into thermal equilibrium. This alignment is subsequently perturbed by the application of a radio frequency pulse. The experiment is concluded by measuring the relaxation time of the sample back to thermal equilibrium, which occurs on the times scale of milliseconds. Furthermore, the rather long time scale of the experiment makes it sensitive to internal rotations, which can blur the difference between unique atoms. For example, it is only at temperatures close to 0 K that one can detect the differences between the hydrogens of a methyl group. In 1995 Augspurger and Dykstra compared NMR shielding in the ground state and lowest excited states in some small organic molecules at the Hartree-Fock (HF) level of approximation [1,2]. Also the technique could be used to characterize the nature of the excited state relative to the ground state. Moreover, in 2008 Karadakov [3,4] reported theoretical excited state NMR parameters for benzene, cyclobutadiene and cyclooctatetraene. These results were primarily used to understand the aromaticity and anti-aromaticity of the ground states and the excited states. Warren et al. [5] mention the triplet excited states in association with the interpretation of the NMR spectra of p-methoxyphenyliminocamphor. Motivated by a possible dependence of NMR spectra on the polarization of incident laser light, they observe a complex pattern of carbon-13 chemical shifts. They found no evidence of such polarization dependence and suggest
E-mail address: [email protected]. https://doi.org/10.1016/j.cplett.2019.03.051 Received 11 March 2018; Received in revised form 24 March 2019; Accepted 25 March 2019 Available online 26 March 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics Letters 724 (2019) 86–89
M. Atsumi
that observed effects are due to differences between ground and excited state chemical shifts. Furthermore, in a 1H and 13C NMR study by Abriata et al. [6] it is proposed that the observed anomalous temperature dependence of a binuclear copper site(CuA) in Thermus thermophilus is explained by contributions from low-lying excited states. The authors attempted experimental identification and characterization of excited states that become populated at room temperature. In this report we will consider the singlet ground state and the singlet excited state NMR calculations on small hydrocarbons. Previous work by Augspurger and Dykstra used the uncorrelated HF level, and it is desirable to obtain more reliable results using correlated methods [1,2].
Table 1 The singlet excited state and the ground state shielding constants (ppm) of the carbon atom and the vertical excitation energy (eV). The shielding constants of the underlined carbon atoms at CH2CCH2 and CH3CHO were presented. PW = present work.
2. Computational details
which the present results also show. Notably, with one exception, the singlet excited states shielding constants were larger than the ground state ones, i.e., the excited states were deshielded, as seen in Table 1. Ramsey's theory [16] can be used for a preliminary analysis of the results. The shielding constants for molecules can be expressed as the sum of diamagnetic and paramagnetic contributions:
The ground state geometries of the molecules were optimized at the Coupled Cluster level with Single, Double, and perturbative Triple excitations (CCSD(T)) included [7]. The program Gaussian 09 was employed [8]. The NMR spectroscopic parameters of the ground state and the singlet excited states were investigated at the State-Specific Complete Active Space Self-Consistent Field (SS-CASSCF) level of theory [9], using the Dalton program package [10] and its MCSCF response module [26,27]. The CASSCF level is suitable both for ground and excited states magnetic property calculations, including magnetic shielding constants. The choice of method is motivated by the fact that electronic excited states typically require a multiconfigurational description and the inclusion of static correlation effects at the CASSCF level should be sufficient for at least qualitative accuracy of NMR properties. For the magnetic property calculations, Gauge-Including Atomic Orbital (GIAO) was used. The active spaces are as follows; CH2CCH2(4e4o), CH2O(10e10o), CH3CHO(12e10o), CH3NH2(10e9o), and CO(10e11o). The basis set 6–311++G(2d,2p) was used for all atoms in the ground state and the singlet excited state. This basis set was chosen since the study by Karadakov reported calculated excited state NMR properties for benzene in good agreement with the experimental results. No symmetry was constrained in the geometry optimization. The ground state geometry was used for the singlet excited state calculations. For the shielding constant calculations, no point group symmetry was used for any of the molecules studied.
Molecule
S1 (PW)
S0 (PW)
S0 (exp) [17]
ΔE (PW)
ΔE (exp)/(state)
CH2CCH2 CH2O CH3CHO CH3CHO CH3NH2 CO
106.0 136.9 184.0 147.0 172.4 179.6
123.9 18.8 164.3 15.6 165.0 16.6
115.1 – 157.1 – 158.2 1.0
6.68 4.63 4.87
6.72(1B2) [20,21] 4.10(1A2) [18] 4.28(1A”) [22]
6.29 11.1
5.64 [23] 10.78(1B2Σ+) [19]
σ = σd + σp We checked the contributions of the diamagnetic and paramagnetic terms to the shielding constants at the excited state (see Table 2). The diamagnetic contribution was similar for all molecules. All carbon atoms had diamagnetic contributions of around 300 ± 20 ppm, except for the contribution 360.1 ppm in CH3CHO, for both the singlet excited states and the ground states. Also the diamagnetic contributions to the shielding constants were remarkably similar to the values at the ground state. On the other hand the paramagnetic contributions showed larger variation between molecules. To rationalize this tendency, note that the diamagnetic contribution depends only on the electron density near each nucleus, which changes fairly little from ground to excited state. The similarity of the molecules we studied may account for the small variation between molecules. As it is well-known that nuclear shielding constants are sensitive to molecular geometry and chemical environment, it is harder to provide a simple explanation for the trend in the paramagnetic contribution. Some insight is provided by the sum-over-states formula in Ramsey's theory. The paramagnetic contribution to the nuclear shielding tensor for the n:th state, Ψn, is
3. Results and discussion
p σαβ =
The geometry of CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO have been optimized at the CCSD(T) level of theory, with 6-311++G(2d,2p) basis sets for all atoms. In the optimized structure of CH2CCH2, both double bond lengths were 1.32 Å. The angle HeCeH was 120°. In CH2O, the optimized molecule exhibited C2v symmetry. The C]O and CeH bond were 1.21 Å and 1.10 Å, respectively. The angle HeCeH was 116.7°. In the optimized structure of CH3CHO, the CeC, C]O, and CeH bonds were 1.51 Å, 1.21 Å, and 1.11 Å. In CH3CHO, the NeH and CeH bonds were 1.01 Å and 1.09 Å, respectively and the single bond CeN was 1.47 Å. In CO, the bond length was 1.14 Å. Compared to the experimental structure, the calculated CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO, the bond lengths differed by less than 0.01 Å [11–15]. They are also in reasonable agreement with the covalent bond lengths [24,25]. Shielding constants, for the ground state and the excited state, were computed at the CASSCF level. All calculated shielding properties refer to the carbon atoms. The theoretical excitation energy (ΔE) was also calculated as the difference between the MCSCF excited state and ground state energy. As seen from Table 1, the calculated ΔE of CH2O and CO is in good agreement with the experimental excitation energies. Furthermore, the shielding constants at the ground state of CH3CHO and CH3NH2 show good agreement with the experimental results. It is well known that the ground state shielding constants are small for CH2O, CH3CHO, and CO,
−1 2c2
∑ m≠n
Λmn , Em − En
where Λmn=〈ψn |Lα |ψm 〉〈ψm |LNβ rN−3 |ψn 〉
+ C. C. where Li is angular momentum relative to the origin, LNi angular momentum relative to nucleus N, and rN is distance to nucleus to N. The Table 2 The calculated diamagnetic and paramagnetic contributions as well as the inverse excitation energy 1/ΔE for the singlet excited states.
87
Diamagnetic
Paramagnetic
Ground state CH2CCH2 CH2O CH3CHO CH3CHO CH3NH2 CO
304.0 309.4 316.1 360.1 301.3 288.1
−180.2 −290.6 −151.8 −344.5 −136.3 −271.5
First excited state CH2CCH2 CH2O CH3CHO CH3CHO CH3NH2 CO
304.5 311.4 317.6 362.8 315.9 282.0
−198.3 −174.6 −133.5 −215.7 −153.7 −102.4
1/ΔE
0.15 0.22 0.21 0.16 0.09
Chemical Physics Letters 724 (2019) 86–89
M. Atsumi
in the carbonyl groups of CH3CHO and CH2O showed large differences between the ground state and the first singlet excited state. The excitation energies were 4.63 eV and 4.87 eV, respectively. For reference, in CH2O, the x-axis was placed along the C]O bond, and the y-axis is the perpendicular to the plane consisted by two hydrogen atoms and oxygen atom, and the z-axis is perpendicular to xy-plane. For CH3CHO, the y-axis was aligned perpendicularly to the plane of two carbon atoms and oxygen atom. With this choice of coordinate axes, the n → π* transitions are from 2pz to 2py at carbon atoms in CH2O and CH3CHO. Looking at the shielding constants in two carbon atoms within the same molecule, CH3CHO is larger than CH3CHO. Next, we investigate the methyl group in CH3NH2. Its excitation energy is 6.29 eV and the transition is mainly organized by the nitrogen atom. The paramagnetic contributions at the carbon nuclei of the methyl groups in CH3CHO and CH3NH2 are substantially smaller (in magnitude) than the corresponding numbers at the other carbon or nitrogen nucleus. In the CO molecule, the excitation with energy 11.1 eV is mainly an n → π* transition from s to p orbital at the carbon. This charge transfer excitation shows a large paramagnetic contribution at the excited state comparing to the ground state. At the singlet first excited state, the charge transfer is associated with a large degree to orbital rotation at the carbon atoms. For CH2CCH2, the geometry optimization and the shielding constants calculation were performed without symmetry constraints. The geometry was to good precision of D2d symmetry. For the active space CAS(4e4o), the shielding constants were symmetric at the edge carbon atoms in the ground state and the excited state. With the larger active spaces that were feasible, we obtained asymmetric nuclear shielding constants at the edge carbons. For this reason, we report results for CAS(4e4o). In the edge carbon atoms, the shielding constants at the excited state is smaller than at the ground state. One contributing reason is that the transition to first excited state in CH2CCH2 is not a magnetically allowed. All excited states in this study turn out to be dominated by oneelectron excitations from the ground state, i.e, two of their natural orbital occupations are close to 1.0, the remaining being close to 2.0 and 0.0. In CH2CCH2, two occupation numbers are 1.4 and 0.6, rather than 1.0. Some carbon atoms in the molecules studied deviated from the rough linear trend because the corresponding excitations are not mainly characterized by either transfer away from the reference atom or a 90 degree rotation of natural orbitals. Augspurger and Dykstra illustrated that useful estimates of chemical shifts can be obtained even with the rough approximation of excitation energies as Hartree-Fock orbital energy differences [1,2]. By decomposing the shielding constant into orbital contributions, they found that most occupied orbitals contribute substantially to both the diamagnetic and paramagnetic terms. Moreover, the paramagnetic shielding constant partially reflected 2p orbital occupations. The conclusion is that electronic excitation cannot consistently be expected to result in either shielding or deshielding. Here we improved on the uncorrelated Hartree-Fock approximations in previous work by calculating the singlet excited state energies for CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO at the CASSCF level of theory, taking into account static correlation effects. Furthermore, we analyzed the excited state shielding constants in terms of Ramsey theory. Compared to previous work, our excitation energies are more reliable and the analysis revealed a large paramagnetic contribution to the excited state NMR shielding constants in the molecules studied. Furthermore in all cases the excited state is quite far from any single configuration wave function. Hence, a correlated treatment is necessary for the excited states.
complex conjugate of the proceeding term is denoted by C.C. The elements of σpαβ (α, β = x, y, z) are the cartesian components of the shielding tensor, which is a non-symmetric tensor. All reported values are isotropic averages, i.e. σp = (1/3)(σpxx + σpyy + σpzz) for the paramagnetic component and similarly for the total shielding constant. Both the ground and first excited state shielding constant share the same first term, Λ01/(E0−E1), in the sum-over-states, save for an opposite sign. Since this term is negative for the ground state, the opposite sign yields a deshielding effect on the first excited state. The matrix elements Λm1 are large when a 90° rotation of the many-electron state Ψm yields large overlap with Ψ1 [16]. For the molecules studied, the states Ψ0, and Ψ1 are built up from similar natural orbitals and differ mainly in occupation numbers. Among the orbitals that differ the most in occupation numbers between Ψ1 and Ψm (m = 0), the pairs that are roughly related by a 90° rotation are likely to give large matrix elements. Another effect that can potentially contribute to the change in σp is charge transfer combined with the distance weighting by rN−3 in the matrix element. When excitations result in charge transfer away from the reference nucleus compared to the ground state, then, ceteris paribus, the distance weighting reduces the magnitude of the paramagnetic contributions. Some orbitals have much larger diamagnetic contributions while others have much larger paramagnetic contributions. However, we focus on analyzing the paramagnetic contributions to these molecule excitations since they dramatically change between the excited states and the ground states. Furthermore and relatedly, the transitions between the ground and excited states studied are magnetically allowed in all systems except CH2CCH2. The results are visualized in Fig. 1, showing a rough linear relationship between the gap to the ground state (ΔE = E1 − E0) and σp. Under idealized simplifying assumptions, an approximately linear relationship between σp and the inverse gap of ΔE is obtained theoretically and comparable relationships have been seen in other studies [28]. In general, the paramagnetic sum-over-states terms are closely related to matrix elements of magnetic dipole (or angular momentum) operators. Hence, electric dipole-forbidden transitions are not prevented from contributing. Turning to a simple molecular orbital picture, with transitions from ground to excited states idealized as a transition from initially occupied to unoccupied orbitals [28], we note that CH2O, CH3CHO, CH3NH2, and CO exemplify n → π* transitions. They involve a large degree of orbital rotation from, say, 2pz to 2py and are magnetically allowed. However, for CH2CCH2 we instead mainly have a π → π* transition, which is not magnetically allowed. As seen Table 1, the paramagnetic contribution at carbon nuclei
4. Conclusions Fig. 1. Data and trend line for the variation of paramagnetic contribution with inverse gap 1/ΔE. The CH2CCH2 excitation is of a different character and is excluded as a data point for the trend line.
We have presented results for the singlet excited state shielding constants that account for correlation effects. For the molecules studied, the singlet excited states are shifted upfield compared to the ground 88
Chemical Physics Letters 724 (2019) 86–89
M. Atsumi
state. This tendency towards deshielding is rationalized using the sumover-states expression for magnetic response. Furthermore, the n → π* transitions in CH2O, CH3CHO, CH3NH2, and CO are magnetically allowed and show a rough linear relationship between paramagnetic shielding contribution and inverse excitation energy, whereas the π → π* transition in CH2CCH2 deviates from the trend.
[9]
[10]
Conflict of interest None declared. Acknowledgement The calculations were performed on resources provided by UNINETT Sigma2 - the National Infrastructure for High Performance Computing and Data Storage in Norway. This work received support from the Research Council of Norway through its Centres of Excellence scheme, project number 262695, and by the Norwegian Supercomputing Program (NOTUR) through a grant of computer time (Grant No. NN4654K). M.A. thanks Dr. E. Tellgren at University of Oslo for the useful discussion.
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Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Research paper
The 13C nuclear magnetic shielding constants at the singlet excited states in CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO
T
Michiko Atsumi Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Norway
H I GH L IG H T S
state NMR shielding constants are computed at the correlated CASSCF level. • Excited excited state shielding constants are deshielded compared to ground state. • Singlet • Large paramagnetic effects are likely due to population of 2p orbitals.
A B S T R A C T
Quantum-chemical calculations of excited state nuclear shielding constants are reported for singlet states in CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO. The multiconfigurational self-consistent field method is used to account for static correlation in the excited states. With one exception, the molecules considered had larger shielding constants in their excited states than in their ground states. This tendency towards deshielding is rationalized using the sum-over-states expression for magnetic response.
1. Introduction Nuclear magnetic resonance (NMR) is presently used to measure the structure of molecules in their electronic ground state. NMR can probe the events on a micro- to nano-second time scale. However, the lifetime of the singlet excited state lies in the picosecond and subpicosecond range. This makes it difficult to measure the molecular structure in the excited states since NMR measurements and the excited state phenomena have different time scales. In this report, we will investigate the NMR spectroscopy of the excited states using quantum chemical methods. Computational techniques will be used to model the electronic wave functions and magnetic properties of ground states and excited states. The differences in structure between ground state and excited state manifest themselves in different values of the nuclear shielding constants. This quantity is sensitive to molecular and electronic structure, and the chemical shift of NMR resonance indicates a change in the electronic structure at the associated nuclei. One advantage of quantum-chemical calculations is that they can address systems and properties that are difficult to access experimentally. The NMR technique utilizes that in the presence of an external magnetic field a nuclear spin state will be aligned with the field. For example, for doublet states (as found in the 1H nuclei) the spin is either parallel or anti-parallel with respect to the external field. The energy difference of two such states, which is normally in the radio-frequency
range, is probed in NMR experiments. However this is not probed in a direct way. Rather, the nuclear spins are exposed to a strong constant magnetic field and allowed to come into thermal equilibrium. This alignment is subsequently perturbed by the application of a radio frequency pulse. The experiment is concluded by measuring the relaxation time of the sample back to thermal equilibrium, which occurs on the times scale of milliseconds. Furthermore, the rather long time scale of the experiment makes it sensitive to internal rotations, which can blur the difference between unique atoms. For example, it is only at temperatures close to 0 K that one can detect the differences between the hydrogens of a methyl group. In 1995 Augspurger and Dykstra compared NMR shielding in the ground state and lowest excited states in some small organic molecules at the Hartree-Fock (HF) level of approximation [1,2]. Also the technique could be used to characterize the nature of the excited state relative to the ground state. Moreover, in 2008 Karadakov [3,4] reported theoretical excited state NMR parameters for benzene, cyclobutadiene and cyclooctatetraene. These results were primarily used to understand the aromaticity and anti-aromaticity of the ground states and the excited states. Warren et al. [5] mention the triplet excited states in association with the interpretation of the NMR spectra of p-methoxyphenyliminocamphor. Motivated by a possible dependence of NMR spectra on the polarization of incident laser light, they observe a complex pattern of carbon-13 chemical shifts. They found no evidence of such polarization dependence and suggest
E-mail address: [email protected]. https://doi.org/10.1016/j.cplett.2019.03.051 Received 11 March 2018; Received in revised form 24 March 2019; Accepted 25 March 2019 Available online 26 March 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.
Chemical Physics Letters 724 (2019) 86–89
M. Atsumi
that observed effects are due to differences between ground and excited state chemical shifts. Furthermore, in a 1H and 13C NMR study by Abriata et al. [6] it is proposed that the observed anomalous temperature dependence of a binuclear copper site(CuA) in Thermus thermophilus is explained by contributions from low-lying excited states. The authors attempted experimental identification and characterization of excited states that become populated at room temperature. In this report we will consider the singlet ground state and the singlet excited state NMR calculations on small hydrocarbons. Previous work by Augspurger and Dykstra used the uncorrelated HF level, and it is desirable to obtain more reliable results using correlated methods [1,2].
Table 1 The singlet excited state and the ground state shielding constants (ppm) of the carbon atom and the vertical excitation energy (eV). The shielding constants of the underlined carbon atoms at CH2CCH2 and CH3CHO were presented. PW = present work.
2. Computational details
which the present results also show. Notably, with one exception, the singlet excited states shielding constants were larger than the ground state ones, i.e., the excited states were deshielded, as seen in Table 1. Ramsey's theory [16] can be used for a preliminary analysis of the results. The shielding constants for molecules can be expressed as the sum of diamagnetic and paramagnetic contributions:
The ground state geometries of the molecules were optimized at the Coupled Cluster level with Single, Double, and perturbative Triple excitations (CCSD(T)) included [7]. The program Gaussian 09 was employed [8]. The NMR spectroscopic parameters of the ground state and the singlet excited states were investigated at the State-Specific Complete Active Space Self-Consistent Field (SS-CASSCF) level of theory [9], using the Dalton program package [10] and its MCSCF response module [26,27]. The CASSCF level is suitable both for ground and excited states magnetic property calculations, including magnetic shielding constants. The choice of method is motivated by the fact that electronic excited states typically require a multiconfigurational description and the inclusion of static correlation effects at the CASSCF level should be sufficient for at least qualitative accuracy of NMR properties. For the magnetic property calculations, Gauge-Including Atomic Orbital (GIAO) was used. The active spaces are as follows; CH2CCH2(4e4o), CH2O(10e10o), CH3CHO(12e10o), CH3NH2(10e9o), and CO(10e11o). The basis set 6–311++G(2d,2p) was used for all atoms in the ground state and the singlet excited state. This basis set was chosen since the study by Karadakov reported calculated excited state NMR properties for benzene in good agreement with the experimental results. No symmetry was constrained in the geometry optimization. The ground state geometry was used for the singlet excited state calculations. For the shielding constant calculations, no point group symmetry was used for any of the molecules studied.
Molecule
S1 (PW)
S0 (PW)
S0 (exp) [17]
ΔE (PW)
ΔE (exp)/(state)
CH2CCH2 CH2O CH3CHO CH3CHO CH3NH2 CO
106.0 136.9 184.0 147.0 172.4 179.6
123.9 18.8 164.3 15.6 165.0 16.6
115.1 – 157.1 – 158.2 1.0
6.68 4.63 4.87
6.72(1B2) [20,21] 4.10(1A2) [18] 4.28(1A”) [22]
6.29 11.1
5.64 [23] 10.78(1B2Σ+) [19]
σ = σd + σp We checked the contributions of the diamagnetic and paramagnetic terms to the shielding constants at the excited state (see Table 2). The diamagnetic contribution was similar for all molecules. All carbon atoms had diamagnetic contributions of around 300 ± 20 ppm, except for the contribution 360.1 ppm in CH3CHO, for both the singlet excited states and the ground states. Also the diamagnetic contributions to the shielding constants were remarkably similar to the values at the ground state. On the other hand the paramagnetic contributions showed larger variation between molecules. To rationalize this tendency, note that the diamagnetic contribution depends only on the electron density near each nucleus, which changes fairly little from ground to excited state. The similarity of the molecules we studied may account for the small variation between molecules. As it is well-known that nuclear shielding constants are sensitive to molecular geometry and chemical environment, it is harder to provide a simple explanation for the trend in the paramagnetic contribution. Some insight is provided by the sum-over-states formula in Ramsey's theory. The paramagnetic contribution to the nuclear shielding tensor for the n:th state, Ψn, is
3. Results and discussion
p σαβ =
The geometry of CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO have been optimized at the CCSD(T) level of theory, with 6-311++G(2d,2p) basis sets for all atoms. In the optimized structure of CH2CCH2, both double bond lengths were 1.32 Å. The angle HeCeH was 120°. In CH2O, the optimized molecule exhibited C2v symmetry. The C]O and CeH bond were 1.21 Å and 1.10 Å, respectively. The angle HeCeH was 116.7°. In the optimized structure of CH3CHO, the CeC, C]O, and CeH bonds were 1.51 Å, 1.21 Å, and 1.11 Å. In CH3CHO, the NeH and CeH bonds were 1.01 Å and 1.09 Å, respectively and the single bond CeN was 1.47 Å. In CO, the bond length was 1.14 Å. Compared to the experimental structure, the calculated CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO, the bond lengths differed by less than 0.01 Å [11–15]. They are also in reasonable agreement with the covalent bond lengths [24,25]. Shielding constants, for the ground state and the excited state, were computed at the CASSCF level. All calculated shielding properties refer to the carbon atoms. The theoretical excitation energy (ΔE) was also calculated as the difference between the MCSCF excited state and ground state energy. As seen from Table 1, the calculated ΔE of CH2O and CO is in good agreement with the experimental excitation energies. Furthermore, the shielding constants at the ground state of CH3CHO and CH3NH2 show good agreement with the experimental results. It is well known that the ground state shielding constants are small for CH2O, CH3CHO, and CO,
−1 2c2
∑ m≠n
Λmn , Em − En
where Λmn=〈ψn |Lα |ψm 〉〈ψm |LNβ rN−3 |ψn 〉
+ C. C. where Li is angular momentum relative to the origin, LNi angular momentum relative to nucleus N, and rN is distance to nucleus to N. The Table 2 The calculated diamagnetic and paramagnetic contributions as well as the inverse excitation energy 1/ΔE for the singlet excited states.
87
Diamagnetic
Paramagnetic
Ground state CH2CCH2 CH2O CH3CHO CH3CHO CH3NH2 CO
304.0 309.4 316.1 360.1 301.3 288.1
−180.2 −290.6 −151.8 −344.5 −136.3 −271.5
First excited state CH2CCH2 CH2O CH3CHO CH3CHO CH3NH2 CO
304.5 311.4 317.6 362.8 315.9 282.0
−198.3 −174.6 −133.5 −215.7 −153.7 −102.4
1/ΔE
0.15 0.22 0.21 0.16 0.09
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in the carbonyl groups of CH3CHO and CH2O showed large differences between the ground state and the first singlet excited state. The excitation energies were 4.63 eV and 4.87 eV, respectively. For reference, in CH2O, the x-axis was placed along the C]O bond, and the y-axis is the perpendicular to the plane consisted by two hydrogen atoms and oxygen atom, and the z-axis is perpendicular to xy-plane. For CH3CHO, the y-axis was aligned perpendicularly to the plane of two carbon atoms and oxygen atom. With this choice of coordinate axes, the n → π* transitions are from 2pz to 2py at carbon atoms in CH2O and CH3CHO. Looking at the shielding constants in two carbon atoms within the same molecule, CH3CHO is larger than CH3CHO. Next, we investigate the methyl group in CH3NH2. Its excitation energy is 6.29 eV and the transition is mainly organized by the nitrogen atom. The paramagnetic contributions at the carbon nuclei of the methyl groups in CH3CHO and CH3NH2 are substantially smaller (in magnitude) than the corresponding numbers at the other carbon or nitrogen nucleus. In the CO molecule, the excitation with energy 11.1 eV is mainly an n → π* transition from s to p orbital at the carbon. This charge transfer excitation shows a large paramagnetic contribution at the excited state comparing to the ground state. At the singlet first excited state, the charge transfer is associated with a large degree to orbital rotation at the carbon atoms. For CH2CCH2, the geometry optimization and the shielding constants calculation were performed without symmetry constraints. The geometry was to good precision of D2d symmetry. For the active space CAS(4e4o), the shielding constants were symmetric at the edge carbon atoms in the ground state and the excited state. With the larger active spaces that were feasible, we obtained asymmetric nuclear shielding constants at the edge carbons. For this reason, we report results for CAS(4e4o). In the edge carbon atoms, the shielding constants at the excited state is smaller than at the ground state. One contributing reason is that the transition to first excited state in CH2CCH2 is not a magnetically allowed. All excited states in this study turn out to be dominated by oneelectron excitations from the ground state, i.e, two of their natural orbital occupations are close to 1.0, the remaining being close to 2.0 and 0.0. In CH2CCH2, two occupation numbers are 1.4 and 0.6, rather than 1.0. Some carbon atoms in the molecules studied deviated from the rough linear trend because the corresponding excitations are not mainly characterized by either transfer away from the reference atom or a 90 degree rotation of natural orbitals. Augspurger and Dykstra illustrated that useful estimates of chemical shifts can be obtained even with the rough approximation of excitation energies as Hartree-Fock orbital energy differences [1,2]. By decomposing the shielding constant into orbital contributions, they found that most occupied orbitals contribute substantially to both the diamagnetic and paramagnetic terms. Moreover, the paramagnetic shielding constant partially reflected 2p orbital occupations. The conclusion is that electronic excitation cannot consistently be expected to result in either shielding or deshielding. Here we improved on the uncorrelated Hartree-Fock approximations in previous work by calculating the singlet excited state energies for CH2CCH2, CH2O, CH3CHO, CH3NH2, and CO at the CASSCF level of theory, taking into account static correlation effects. Furthermore, we analyzed the excited state shielding constants in terms of Ramsey theory. Compared to previous work, our excitation energies are more reliable and the analysis revealed a large paramagnetic contribution to the excited state NMR shielding constants in the molecules studied. Furthermore in all cases the excited state is quite far from any single configuration wave function. Hence, a correlated treatment is necessary for the excited states.
complex conjugate of the proceeding term is denoted by C.C. The elements of σpαβ (α, β = x, y, z) are the cartesian components of the shielding tensor, which is a non-symmetric tensor. All reported values are isotropic averages, i.e. σp = (1/3)(σpxx + σpyy + σpzz) for the paramagnetic component and similarly for the total shielding constant. Both the ground and first excited state shielding constant share the same first term, Λ01/(E0−E1), in the sum-over-states, save for an opposite sign. Since this term is negative for the ground state, the opposite sign yields a deshielding effect on the first excited state. The matrix elements Λm1 are large when a 90° rotation of the many-electron state Ψm yields large overlap with Ψ1 [16]. For the molecules studied, the states Ψ0, and Ψ1 are built up from similar natural orbitals and differ mainly in occupation numbers. Among the orbitals that differ the most in occupation numbers between Ψ1 and Ψm (m = 0), the pairs that are roughly related by a 90° rotation are likely to give large matrix elements. Another effect that can potentially contribute to the change in σp is charge transfer combined with the distance weighting by rN−3 in the matrix element. When excitations result in charge transfer away from the reference nucleus compared to the ground state, then, ceteris paribus, the distance weighting reduces the magnitude of the paramagnetic contributions. Some orbitals have much larger diamagnetic contributions while others have much larger paramagnetic contributions. However, we focus on analyzing the paramagnetic contributions to these molecule excitations since they dramatically change between the excited states and the ground states. Furthermore and relatedly, the transitions between the ground and excited states studied are magnetically allowed in all systems except CH2CCH2. The results are visualized in Fig. 1, showing a rough linear relationship between the gap to the ground state (ΔE = E1 − E0) and σp. Under idealized simplifying assumptions, an approximately linear relationship between σp and the inverse gap of ΔE is obtained theoretically and comparable relationships have been seen in other studies [28]. In general, the paramagnetic sum-over-states terms are closely related to matrix elements of magnetic dipole (or angular momentum) operators. Hence, electric dipole-forbidden transitions are not prevented from contributing. Turning to a simple molecular orbital picture, with transitions from ground to excited states idealized as a transition from initially occupied to unoccupied orbitals [28], we note that CH2O, CH3CHO, CH3NH2, and CO exemplify n → π* transitions. They involve a large degree of orbital rotation from, say, 2pz to 2py and are magnetically allowed. However, for CH2CCH2 we instead mainly have a π → π* transition, which is not magnetically allowed. As seen Table 1, the paramagnetic contribution at carbon nuclei
4. Conclusions Fig. 1. Data and trend line for the variation of paramagnetic contribution with inverse gap 1/ΔE. The CH2CCH2 excitation is of a different character and is excluded as a data point for the trend line.
We have presented results for the singlet excited state shielding constants that account for correlation effects. For the molecules studied, the singlet excited states are shifted upfield compared to the ground 88
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state. This tendency towards deshielding is rationalized using the sumover-states expression for magnetic response. Furthermore, the n → π* transitions in CH2O, CH3CHO, CH3NH2, and CO are magnetically allowed and show a rough linear relationship between paramagnetic shielding contribution and inverse excitation energy, whereas the π → π* transition in CH2CCH2 deviates from the trend.
[9]
[10]
Conflict of interest None declared. Acknowledgement The calculations were performed on resources provided by UNINETT Sigma2 - the National Infrastructure for High Performance Computing and Data Storage in Norway. This work received support from the Research Council of Norway through its Centres of Excellence scheme, project number 262695, and by the Norwegian Supercomputing Program (NOTUR) through a grant of computer time (Grant No. NN4654K). M.A. thanks Dr. E. Tellgren at University of Oslo for the useful discussion.
[11]
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