Properties of the Nitrogen-Vacancy
Point Defect in Diamond
A. S. ZYUBIN,1A. M. MEBEL,2M. HAYASHI,3H. C. CHANG,4S. H. LIN5
1
Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia
2
Department of Chemistry and Biochemistry, Florida International University, Miami, Florida 33199
3
Center for Condensed Matter Science, National Taiwan University, Roosevelt Rd., Sec. 4, Taipei 10764, Taiwan
4Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166,
Taipei 106, Taiwan
5
Department of Applied Chemistry and Institute of Molecular Sciences, National Chiao-Tung University, Hsin-chu, Taiwan
Received 12 November 2007; Revised 11 March 2008; Accepted 28 April 2008 DOI 10.1002/jcc.21042
Published online 11 June 2008 in Wiley InterScience (www.interscience.wiley.com).
Abstract: Quantum chemical calculations of geometric and electronic structure and vertical transition energies for several low-lying excited states of the neutral and negatively charged nitrogen-vacancy point defect in diamond (NV0and NV2) have been performed employing various theoretical methods and basis sets and using finite model NCnHm clusters. Unpaired electrons in the ground doublet state of NV0 and triplet state of NV2 are found to be
localized mainly on three carbon atoms around the vacancy and the electronic density on the nitrogen and rest of C atoms is only weakly disturbed. The lowest excited states involve different electronic distributions on molecular orbitals localized close to the vacancy and their wave functions exhibit a strong multireference character with signif-icant contributions from diffuse functions. CASSCF calculations underestimate excitation energies for the anionic defect and overestimate those for the neutral system. The inclusion of dynamic electronic correlation at the CASPT2 level leads to a reasonable agreement (within 0.25 eV) of the calculated transition energy to the lowest excited state with experiment for both systems. Several excited states for NV2are found in the energy range of 2–3 eV, but only for the 13E and 53E states the excitation probabilities from the ground state are significant, with the first absorption band calculated at1.9 eV and the second lying 0.8–1 eV higher in energy than the first one. For NV0, we predict the following order of electronic states: 12E (0.0), 12A
2 (2.4 eV), 22E (2.7–2.8 eV), 12A1, 32E (3.2 eV and
higher).
q 2008 Wiley Periodicals, Inc. J Comput Chem 30: 119–131, 2009
Key words: diamond; photoabsorption; ab initio calculations; excited electronic states
Introduction
Diamond and diamond-like materials are finding increasing applications in various areas of modern technology because of their unique properties such as high thermal conductivity, excep-tional hardness, high damage threshold of the host crystal, and chemical and thermal stability. Diamonds contain a variety of color centers owing to the presence of impurity atoms and va-cancy complexes. Up to date, based on the absorption and lumi-nescence spectra of diamond, more than 500 optical centers are
known, which are mostly related to the presence of defects.1–3 Approximately half of them are believed to be impurity related. Properties of these centers are of great interest and have been studied for more than 30 years because of the proposed
Contract/grant sponsors: Academia Sinica, National Science Council of Taiwan (R.O.C), Florida International University
applications of diamond as a laser source and high-efficiency photodetector for the UV radiation.4–7Nitrogen is the dominant impurity in the bulk of natural and synthetic diamond.8–10 It is usually present in the form of substitution atoms (sometimes called type C nitrogen), which may be isolated or be present in aggregates. All synthetic diamonds, either HPHT (High Temper-ature-High Pressure) or CVD (chemical vapor deposition), con-tain type C nitrogen, except those from which nitrogen has been carefully excluded. C nitrogen gives rise to a characteristic IR spectrum with a broad peak around 1130 cm21,11a sharp local mode peak at 1344 cm21, and it can also serve as an electron donor with ionization energy of 1.7–2 eV.12The presence of C nitrogen gives rise to the P1 EPR signal.13 A single nitrogen atom impurity can be ionized, and then the N1ion generates an IR peak at 1332 cm21.14
In natural diamond, vacancies are present only after irradiation. The ground state of the neutral vacancy (V0) is singlet; four elec-trons used to be attached to the removed atom form weak, stretched CC bonds. EPR signals have been observed associated with V2(S1)15and with an excited state of V0.16The optical band GR1 with zero phonone line (ZPL) at 1.673 eV has been attributed to V0, whereas ND1 with ZPL at 3.150 eV has been assigned to the negatively charged vacancy V2.8The migration energy for the vacancy is2.3 6 0.2 eV, and at the temperature of 6008C neu-tral vacancies begin to migrate17 and to readily form complexes with impurities creating more stable optically active defects.
Several centers incorporating nitrogen are known to be ther-mally stable and to exhibit high quantum efficiency up to tem-peratures above 500 K. These centers are denoted as NV (N&V), H3 (2N&V), H2 (2N&V2), and N3 (3N&V). They are respectively formed by combining the vacancy and one, two, or three N atoms and have C3v and C2v point symmetry. The
exceptional properties of these diamond color centers, together with their high thermal conductivity, outstanding hardness, and high damage threshold of the host crystal caused substantial in-terest of researches.1,3
Nitrogen-vacancy (N-V) defects exhibit a particularly strong electronic transition, allowing detection of individual centers opti-cally.6This could be regarded as a good example of an individual quantum object with potential applications to the quantum infor-mation processing. These defects possess a remarkable photo and chemical stability, and therefore they have been proposed as light sources for high-resolution scanning-probe microscopy and quan-tum cryptography5–7,18 where a stable single-photon source with high quantum yield is required. Recently, nanosized diamond powders containing NV-defects were proposed as cellular biomarkers.19 The NV defect center in diamond has been exten-sively investigated both experimentally1,3,5–14,18,20–37and theoreti-cally.38–42According to available experimental data,3,26the
neu-tral NV defect can capture an electron with formation of a nega-tively charged center. The ground state of this NV2 center is
3
A2, and the first triplet excited state is3E. Maxima of the
pho-toadsorption (PA) and photoluminescence (PL) bands have been observed near 2.15 and 1.75 eV.25So as the Stokes shift is mod-erate in this case, geometric parameters of the3A2and 3E states
do not differ significantly and vibrational wave functions withn 5 0 for these electronic states overlap. This makes possible ob-servation of a direct transition between the ground vibrational
states. The corresponding ZPL at 1.945 eV exhibits a strong in-homogeneous broadening attributed to a large strain variation in the3E state.25
An electron from NV2 can be removed with formation of the NV0 defect. In this case, the ZPL energy is higher, 2.16 eV.28,29
The irradiation of NV2 centers with different sources (2.21–2.43, 2.53–2.76, and 3.26–3.76 eV photons) causes significant variations in the PL band,32which can be due
to excitations of higher electronic states of NV2, involvement of NV0 centers into the process, or both. Hence, detailed informa-tion about properties of several excited states of the NV defects is required; and high-level quantum chemical calculations would be useful for this purpose.
Upon optical excitation, the saturation signal of the NV2 center is more than an order of magnitude lower that one would expect from a two-level system. Weak saturation of fluorescence intensity can be explained by the presence of a metastable sin-glet state (1A). However, ensemble experiments give only small indication of this state to be present. More direct evidence for the metastable state (the shelving effect) was obtained in single center experiments.3The shelving effect explains the decrease of the NV2luminescence beyond a detectable level at temperatures below 80 K. On the basis of the temperature dependence of the single center fluorescence intensity, the energy difference between the 3E and 1A states was determined as 300 cm21. Such a feature of the electronic structure creates some problems for using this defect, especially at low temperatures. This bottle-neck can be overcome by employing a repumping laser, but the deshelving procedure leads to strong broadening of spectral lines.3Because of the small energy gap between the 3E and1A states, their relative position can be exchanged by modifying the defect vicinity or the defect itself. Another approach is to use a similar defect but with another structure and symmetry. Quan-tum chemical calculations would be also useful for testing dif-ferent possibilities.
Theoretical calculations of the properties of NV2available in the literature were performed at the DFT level with local spin density approximation (LSDA) using a model cluster including about 70 atoms with a small Gaussian basis set.41Another group employed self-consistent spin-polarized calculations within a generalized gradient approximation (GGA) using the full-poten-tial linearized augmented plane wave (LAPW) method with peri-odical boundary conditions42and the analysis of electronic states was carried out on the basis of Kohn-Sham eigenvalues. In ref. 41 a more accurate evaluation of optical transition energies was made by using the Slater transition method, in which the transi-tion energy is calculated as a difference between first derivatives of the energy for the special state containing half electron at the donating and accepting orbitals with respect to occupation num-bers of these orbitals. Similar approaches are widely used for modeling of other point defects in diamond.43–54
The theoretical results for the NV2 center are in reasonable agreement with the experimental data, however, other publica-tions offered different interpretation of the electronic structure of this defect.36,38They cast a doubt on the thermal stability of the negatively charged defect and construe the observed properties exclusively based on the neutral state and the appearance of the Jahn-Teller effect. This triggered a discussion concerning
vari-ous explanations of the available experimental results and possi-ble electronic structure of the defect responsipossi-ble for the 1.945 eV ZPL band.27,39,40 In our opinion, the model from refs. 36 and 38 is not solidly substantiated. However, the approach used in ref. 41 is also not very reliable. First, a small basis set was used without polarization and diffuse functions, which usually leads to significant overestimation of excitation energies. Sec-ond, the DFT approach sometimes fails for open-shell point defects55,56 underestimating transition energies. Therefore, for some systems reasonable agreement between the calculated and experimental results may be due to accidental error cancellation; in other similar cases the accuracy of theoretical results may be much lower. Hence, it is worthwhile to perform additional theo-retical calculations of the NV0 and NV2 defects applying more sophisticated methods based on the multireference approach.
The studies (especially theoretical) of the defects containing two and three N atoms (N3, H3, H2) are scarce, despite the fact that these centers also have remarkable properties. It was pro-posed that N3 with ZPL of 2.8 eV has electronic structure qualitatively similar to that of NV2: the ground state of the A-type, E and A excited states with a small energy difference where the A state is slightly lower.1 Theoretical modeling47 at the LSDA level did not reveal such states, and instead an excita-tion to a shallow a1trap localized outside the vacancy (a charge
transfer transition) was suggested. Clearly, calculations with larger basis sets and utilizing more sophisticated theoretical methods would be informative. For the H3 defect with ZPL of 2.33 or 2.46 eV1,31
additional higher states were found, possi-bly with involvement of the negatively charged state with ZPL of1.26 eV for the first transition.31Quantum chemical calcula-tions would be useful for elucidating the electronic structure and interpretation of the available data for these centers.
A vacancy in diamond can be combined with other impur-ities. Admixture atoms larger than C or N (Si and P) are unable to occupy the same position as the substituted C atom and are usually located approximately in the center of a di-vacancy giving rise to electronic structure with lower ZPL values of 1.7 eV.41,48
Thus, combination of substitution atoms and a va-cancy results in several weakened CC bonds or unpaired elec-trons and creates a complex electronic system with electron tran-sitions in IR, UV, and visible ranges. Variations of the type and the number of impurity atoms or functional groups give an op-portunity to create optically active centers with different desira-ble properties. For selective production of quantum dots with required characteristics it is necessary to elucidate how the admixtures influence the defect properties and quantum chemical modeling can be a useful tool for this task. Therefore, the stud-ies of optical propertstud-ies of local defects in diamond represent a broad field for applications of quantum chemical calculations. As a first step in this direction, the goal of this article is model-ing of photoadsorption properties of the nitrogen-vacancy defect. However, one can expect that defects in diamond possess complicated, multideterminant wave functions, especially for excited electronic states. Hence, the modeling should be based on multireference methods, such as, for instance, CASSCF (MCSCF),57,58MRDCI,59MRCI,58,60,61and CASPT2.62,63So as it is feasible, from the point of view of computational demands, to include only few electrons and molecular orbitals into an
active space of CASSCF calculations, the account for electron correlation in this approach is very limited, as dynamic correla-tion is not included. With proper seleccorrela-tion of active space it can provide qualitatively correct description of the wave function and reasonable values of geometry parameters, but for energetic properties its accuracy is not sufficient. More sophisticated MRDCI and MRCI methods give accurate results for small mol-ecules, but for moderate-size systems it is not feasible to take into account electron correlation for all valence electrons, and therefore these approaches are also limited. At present, the most suitable approach for multireference systems with several tens of valence electrons is CASPT2,62,63where the account of electron correlation for all valence electrons is performed on the basis of a CASSCF wave function utilizing the second order perturbation theory. The CASPT2 method is mainly used in this work for calculations of excitation energies for NV-defects in diamonds.
The Choice of Model Clusters and Calculation Details
Quantum chemical calculations of a diamond fragment and the NV-defect were carried out using three types of model clusters with termination of broken boundary bonds by H atoms, C21H28,
C35H36, and C51H52 (see Fig. 1). In the first system, the central
atom is surrounded by two carbon shells forming a C17system,
but in this case it is impossible to terminate all boundary bonds by H atoms, as some centers become very close to each other. This difficulty can be overcome by adding four C atoms with the same bond distances and bond angles as those in diamond but with torsion angles shifted by 608: see the C21H28 cluster,
where boundary C atoms have two CH bonds each. Another, but more expensive way to solve this problem is to use a larger C35H36cluster, where C atoms from the second shell have one
CH bond each. In the C51H52cluster, all C atoms from the
sec-ond shell have CC bsec-onds only; to avoid crowding of boundary H we used the same approach as for the C21H28 system.
Addi-tionally, we also used a small C5H12 model cluster for testing
purposes.
Geometry optimization of the C51H52cluster was performed
at the HF and B3LYP levels with the standard 6-31G* basis set for central 17 atoms and 6-31G for all remaining atoms. Outer C atoms were kept frozen in the crystalline positions, and coor-dinates of the C atoms from the first and second shells and all H atoms were fully optimized. The same approach was used for C35H36 and additionally we carried out full optimization with
the 6-31G* basis set. Geometries of smaller clusters were fully optimized at the HF, B3LYP, and MP2 levels of theory. The GAUSSIAN 03 program package64 was utilized for these calcu-lations. The calculated CC bond lengths are in good agree-ment with the experiagree-mental value for diamond for all models in use, but the most accurate results with deviations from experi-ment of only0.01 A˚ were obtained at the HF and MP2 levels; at B3LYP the CC distances are slightly overestimated (see Table 1). The influence of the cluster size on the calculated geo-metric parameters appeared to be weak.
Modeling of the NV-defect was based on the three clusters discussed earlier. The vacancy was placed in the center of the
cluster and one C atom from the inner shell was replaced by N (see Fig. 1). The outer atoms were kept frozen in the positions of the defect-less system, and atomic coordinates for the first and second shells (four and 12 C atoms, respectively) were opti-mized within Cssymmetry constrains to allow for possible
Jahn-Teller distortion. So as the NV defect can have a negative charge, the basis set used for the calculations of the initial clus-ters was augmented by diffuse s- and p-functions as in the 6-311G* basis set for atoms from the first shell and s-functions for atoms from the second shell. Three electronic states were calculated: negatively charged triplet (nt), neutral doublet (d), and neutral quartet (q). Calculations were performed at the HF and B3LYP level and no principal differences between elec-tronic structures obtained by these two methods were found.
The electron density of HF boundary molecular orbitals (MOs) for the smallest (C3NVH12) and middle-size (C33NVH36)
model clusters are plotted in Figures 2 and 3, respectively. In both systems, the compositions and shapes of appropriate MOs are similar, but the picture for the small cluster is more simple and clear. The lowest MO from this group (HOMO-1) corre-sponds to the nitrogen lone pair; it is doubly occupied in all considered states. The next MO (HOMO) is mainly constructed from broken bonds of vacancy-related C atoms; it has weak CC bonding and CN antibonding characters. This MO is doubly occupied in the negative triplet and neutral doublet states, but singly occupied in the quartet state. The next two MOs (SOMO-A0 and SOMO-A@) are components of the doubly degenerate E orbital with a nonbonding CC character. These MOs are singly occupied in the nt and q states and share one electron in the d state. The unoccupied (empty) MOs are very diffuse; LUMO is similar to a Rydberg-type s atomic orbital (AO) and LUMO 11 is reminiscent of a p-AO. However, one Figure 1. Model clusters used for simulations of a diamond fragment and the nitrogen-vacancy (NV)
has to keep in mind that when these orbitals are involved in multideterminant CASSCF wave functions, their shapes can sig-nificantly change.
In the quartet state, three unpaired electrons are located mainly at inner C atoms with broken bonds and occupy orbitals of e- and a-types (a1e1e1), as the actual symmetry of the system is C3v. As compared to the defect-less system, atoms of the
inner shell are shifted from the center by 0.20–0.25 A˚ in the small cluster and by 0.10–0.12 A˚ in the biggest one; the shift is more significant for C atoms than for the nitrogen. Nevertheless, distances between chemically bonded atoms are similar with deviations of only 0.01–0.03 A˚ (Table 2). Atoms from the sec-ond layer exhibit only a small shift from the center,0.01 A˚. In
all the models, interatomic bonds between threefold coordinated atoms and their nearest neighbors (NC0 and CC@) are 0.05– 0.07 A˚ shorter than the CC distance in the defect-less cluster.
In the doublet state the occupancy of different components of degenerate e-orbitals is not equivalent, which leads to Jahn-Teller distortion. However, in the system under investigation it concerns mainly electronic distribution, especially spin density (Table 3), so as the geometric distortion is very weak; distances from the center are split by 0.01–0.02 A˚ , whereas the splitting of bond lengths is practically zero (Table 2). The main trends in geometry changes are approximately the same as for the quartet state.
In the negative triplet state the wave function has C3v
sym-metry and the additional electron occupies the a MO giving rise
Table 1. Optimized Geometric Parameters of Model Clusters Without Defects.
R1(HF) R1(B3LYP) R1(MP2) R2(HF) R2(B3LYP) R2(MP2) C5H12 a 1.535 1.540 1.530 – – – C21H28 a 1.569 1.576 1.563 1.564 1.570 1.560 C35H36 1.557 1.565, 1.564a – 1.547 1.540, 1.551a – C51H52 1.548 1.548 – 1.561 1.563 –
R1and R2are CC distances in the central fragment containing 17 carbon atoms and in the second layer
surround-ing the central fragment, respectively. The experimental CC bond length in diamond is 1.546 A˚.
aFull geometry optimization was carried out with the 6-31G* basis set. Otherwise, the 6-31G* basis set was used for
seventeen central C atoms and 6-31G for all others atoms. The positions of the central atoms were optimized and the coordinates of outer (boundary) carbons were frozen in crystalline positions.
Figure 2. Electron density plot of boundary orbitals for the smallest C3NV cluster obtained from HF
to the a2e1e1electronic configuration. The main change in geom-etry compared to the neutral states is a larger shift of N and smaller shifts of C atoms, apparently due to the effect of the
additional electronic density, which is redistributed at the broken bonds of the defect (Table 2). At the HF level, the negative tri-plet state is less stable than the neutral doublet and quartet, how-Figure 3. Electron density plot of boundary orbitals for the middle-size C33NV cluster obtained from
HF calculations.
Table 2. Optimized Interatomic Distances R(AB), in A˚ , and Relative Energies Er(in eV) for Model Clusters
Containing the NV-Defect in Different Electronic States. System, method, spin state Er R (VN) R (VC0 , @) R (NC1,2) R (C0 , @C0, @1,2) R (VC01,2) R (VC@1,2) A1, HF, nt 1.97 1.85 1.78 1.47 1.50, 1.51 2.52 2.53–2.55 A1, HF, d 0.00 1.78 1.80, 1.82 1.48 1.51 2.52 2.54–2.56 A1, HF, q 0.17 1.77 1.83 1.48 1.51 2.52 2.55–2.56 A1, B3LYP, nt 0.00 1.88 1.79 1.47 1.50 2.54 2.54 A1, B3LYP, d 0.07 1.82 1.79, 1.82 1.48 1.50–1.51 2.54 2.55 A1 B3LYP, q 0.36 1.78 1.85 1.49 1.50 2.54 2.55 A2, HF, nt 2.14 1.70 1.67 1.48 1.50, 1.51 2.50 2.51, 2.52 A2, HF, d 0.00 1.66 1.71, 1.72 1.49 1.50–1.51 2.50–2.51 2.52–2.53 A2, HF, q 0.26 1.65 1.74 1.50 1.50, 1.51 2.50 2.53 A3, HF, nt 2.19 1.68 1.61 1.49 1.52, 1.54 2.53 2.53, 2.54 A3, HF, d 0.00 1.64 1.66, 1.68 1.51 1.52–1.53 2.53 2.53–2.54 A3, HF, q 0.31 1.63 1.67 1.51 1.52, 1.53 2.53 2.54
nt, d, and q denote negative triplet, neutral doublet, and quartet spin states, respectively. A1, A2, and A3 designate C19NVH28, C33NVH36, and C49NVH52model clusters. The 6-311G* basis set was used for central atoms 1–4 (the
first layer around the vacancy), 6-31G* with a diffuse s-AO at atoms 5–16 (the second layer around atoms 1–4), and 6-31G for all other atoms.
ever, the inclusion of electron correlation stabilizes the nega-tively charged state.
For negative triplet and neutral doublet, geometric parameters were additionally optimized at the CASSCF level with two active spaces, including 8 or 4 active electrons distributed on 11 MOs (Table 3). For the NV21defect the resulting geome-try is slightly distorted from C3v symmetry, probably due to a
limited active space; the inclusion of extra vacant orbitals into the active space, from (8,11) to (4,11), results in the geometry closer to the C3v-symmetric structure. Nevertheless, the
CASSCF optimized geometric parameters are similar with the two active spaces and very close to those obtained at B3LYP (compare Tables 2 and 3). For neutral doublet (NV0) the distor-tion of geometry from C3vsymmetry is more substantial; it
con-cerns mainly the distances between the vacancy center and va-cancy-related C atoms, for which the differences reach 0.02– 0.04 A˚ . Depending on the optimized state, 12A0 or 12A@, the deviations have opposite signs and almost equal absolute values, and the averaged geometry is very close to C3vsymmetry (Table
3). So as one can expect equal populations of both 12A0 and 12A@ components in the degenerate 12E state, the geometry of NV0in this state will probably have C3vsymmetry.
The results obtained with clusters of different size are simi-lar; the most significant difference concerns the shift of inner shell atoms from the center in the smallest cluster, but bond dis-tances are much closer to each other. For the NVC49H52 and
NVC33H36systems all characteristics are very similar. We tried
to use the small cluster with frozen atoms of the second shell, where the atomic coordinates were taken from NVC49H52, but
the results did not change significantly. Additionally, we tested the effect of the basis set extension by adding diffuse p-func-tions to the second layer atoms. The results were almost identi-cal to those obtained with the initial basis set.
There is no substantial difference between geometric parame-ters calculated within the HF and B3LYP methods, however, rel-ative energies of the electronic states were significantly affected. At the B3LYP level negative triplet is the most stable, but the difference between the nt and d states is small (Table 2) and it remains small at MP2, 0.0 and 0.2 eV, respectively. At the nt
geometry (vertical ionization) the calculated energy differences between nt, d and q states are slightly higher, 0.4 and 0.8 eV. The basis set extension to 3111G* for the first shell and 6-311G* for the second one gives the energy differences of 0.5, 0.9 eV at B3LYP and 0.3, 0.7 eV at MP2. With the large NVC49H52cluster, the relative stabilities did not change
signifi-cantly, 0.1 and 0.7 eV at B3LYP.
Mulliken spin distributions at the HF and B3LYP levels are qualitatively similar, but there are some quantitative differences. Unpaired electrons are located mainly at the three inner C atoms of the defect, producing spin polarization of electron density at the atoms linked to this group; the spin density at N and at N-bound C atoms remains weakly disturbed. So as the determina-tion of Mulliken charges and densities is rather arbitrary, espe-cially for basis sets containing diffuse functions, we also calcu-lated isotropic Fermi contact couplings at the14N and13C nuclei as a measure of spin distribution. Considering this property, the qualitative picture remains the same: the largest constants are found for the threefold coordinated C atoms, much smaller val-ues (10 times as small) and with opposite signs––for their neighbors, and nearly zero––at nitrogen and N-bound C nuclei. These results indicate that the nitrogen-related electron distribu-tion is rather rigid and is not involved in electronic rearrange-ment with variations of the electronic state of this defect.
Excited States of the NV2 Defect
In NV2, three hybrid orbitals of the C atoms around the vacancy form a1and e orbitals and the ground state of this system (3A2)
has the a12e1e1electronic configuration with two unpaired
elec-trons occupying the degenerate e orbitals. Additionally, the N lone pair (LP) corresponds to a lower doubly occupied a1 MO.
The first triplet excited state may be formed by the a1? e
elec-tronic transition at the vacancy-related C atoms and could be of the3E type. Formation of higher excited states can be achieved with the involvement of the N lone pair (the a12? e1transition)
or e1and unoccupied MOs (LUMOs). To test this schematic pic-ture and to estimate the influence of the basis set quality and
Table 3. Interatomic Distances R(AB), in A˚ , for the Ground States of the NV21and NV0Defects Optimized
at the CASSCF level.
R(AB) VN VC0 VC@ NC1 NC2 C0C01 C0C02 C@C@1 C@C@2 13A@, (8,11)a 1.887 1.770 1.764 1.47 1.47 1.51 1.52 1.53 1.52 13A@, (4,11)a 1.886 1.765 1.769 1.47 1.47 1.51 1.52 1.51 1.52 12A0, (8,11)a 1.844 1.825 1.781 1.48 1.48 1.52 1.52 1.53 1.52 12A@, (8,11)a 1.843 1.774 1.806 1.48 1.48 1.52 1.53 1.52 1.52 Averagedb 1.844 1.800 1.793 1.48 1.48 1.52 1.52 1.52 1.52 12A0, (4,11)a 1.829 1.822 1.777 1.48 1.48 1.52 1.52 1.53 1.52 12A@, (4,11)a 1.829 1.771 1.804 1.48 1.48 1.52 1.53 1.52 1.52 Averagedb 1.829 1.796 1.790 1.48 1.48 1.52 1.52 1.52 1.52
The A1 model cluster was used in the calculations. The basis sets were 6-311G* for N, C0, C@ atoms, 6-31G* 1 s(0.0438) for C atoms 5–16 in the second layer, and 6-31G basis set for all other atoms.
a
Active space used in CASSCF calculations.
b
model cluster size on the calculated excitation energies, we ini-tially performed TD-DFT(B3LYP) calculations of vertical exci-tation energies for several lowest tripled states (Table 4). The qualitative picture according to the aforementioned scheme was reproduced only with the smallest 6-31G basis set for the C19NV model cluster (A1). The first excited state (13E,
1.8 eV) is then formed by the a12? e1 transition, the LP(N)
? e1
excitation creates the 33E state (4.3 eV), and all other available states in the 3.6–4.5 eV range are formed mainly by excitations from the singly occupied e1 to empty MOs. How-ever, the basis set extension by including polarization and dif-fuse functions changes this picture significantly. The addition of diffuse functions on the vacancy-related N and C atoms in the first layer around V leads to a moderate decrease of the 13E energy by 0.15 eV, but the other states fall down drastically, by about 1 eV. The diffuse functions on C atoms from the ond layer cause a further energy decrease, especially for the sec-ond and higher excited states. The effect of polarization d-func-tions is weaker and the difference between the calculated values with the 6-3111G* and 6-311G* basis sets is rather small. With our best basis set (atoms 1–4: 6-3111G*; 5-16: 6-311G*; 17-20: 6-311G), the relative energy of the 13
E state,1.4 eV, is significantly underestimated and the next higher excited states lie in the close range of 1.71–2.44 eV (Table 4). The 13E state
is formed mainly by the a12 ? e1 transition, but the
contribu-tions of e1 ? LUMOs are also substantial; the other states are formed mainly by the e1 ? LUMOs excitations, while the LP(N) ? e1 excitation is not involved. With the larger C33NVH36 cluster, the relative energy of the first excited state
remains nearly unchanged, but the other states are lower in energy than those for the smaller cluster (Table 4). So as all these states are formed with contribution of diffuse (Rydberg-type) orbitals, their excitation energies should be largely under-estimated at the TD-DFT level. In all cases the first excitation exhibits the largest oscillator strength and the other transitions are much weaker (Table 4). Certainly, we have to keep in mind
imperfection of this theoretical method, which usually signifi-cantly underestimates excitation energies for states with diffuse electronic distribution, and so the effect of diffuse functions might be overestimated in the TD-DFT calculations. Neverthe-less, these calculations demonstrate that the influence of diffuse functions is essential, and they need to be included at least for the vacancy-related atoms to adequately reproduce the transition energy even for the lowest excited states. In all cases, the excited states are formed with contributions from several one-electron transitions with significant weights. This indicates that in this system the excited states have multideterminant wave functions. It should be also noted that the strong influence of diffuse functions on the calculated excitation energies in cluster models may point to the fact that the excited states extend beyond the cluster border and may have nearly-conduction band character, in which case such cluster is no longer appropriate to describe the corresponding state.
CASSCF calculations of excited states were performed using two model clusters, small C3NH12 and moderate C19NVH28. The
small cluster was used for the active space selection and a qualita-tive analysis of the results and then the excitation energies were refined with the moderate model cluster. The calculations with dif-ferent active spaces demonstrated that in the lowest excited states four electrons from the threefold-coordinated C atoms are local-ized close to the center of the cluster at the vacancy position. Con-tributions from the electrons involved in bonding of these atoms with the rest of the bulk (CC@ bonds) and from the N-related elec-trons, both from the N lone pair and NC0 bonds, are small; the occupation numbers of the corresponding MOs included into the active space remain to be close to 2.0. Significant deviations from 2.0 or 0.0 are found for the occupation numbers of the MOs, cor-responding to highest occupied (a12e1e1) and lowest unoccupied
orbitals of the ground state determinant, which are composed from the AOs of threefold coordinated C atoms and diffuse functions. This means that the lowest excitations of the NV-defect are formed by electronic redistribution at the C atoms with broken
Table 4. Relative Energies (eV) and Oscillator Strengths (in bold) for Lowest Excited States of the NV2 Defect Calculated at the Ground Triplet State Geometries.
System, method 13A2 1 1 E 11A1 1 3 E 21E 23E 23A2 A0, CIS(D) 0.0 – – 1.90, 0.155 – 3.20, 0.059 1.69, 0.023 A0, CAS(14,14) 0.0 0.28 0.75 0.96, 0.152 1.07 1.24, 0.070 – A0, MRCI(14,12)a 0.0 0.370.42 1.111.17 1.351.43 1.361.43 1.561.52 – A0, CAS(8,15) 0.0 0.38 0.96 1.14, 0.134 1.20 1.37, 0.072 – A0, MRCI(8,12)a 0.0 0.43 1.16 1.411.46 1.47 1.611.65 – A1, TD-B3LYP 0.0 – – 1.52, 0.088 – 2.21, 0.014 2.13, 0.001 A1, CAS(8,11) 0.0 0.44 1.00 0.98, 0.188 1.13 1.22, 0.097 – A1, MRCI(8,10)a 0.0 0.450.50 1.221.23 1.301.36 1.371.37 1.661.61 – A1, CAS(8,11)b 0.0 – – 1.15, 0.137 – – –
A1, MRCI(8,10)a,b 0.0 – – 1.411.46 – – –
A2, TD-B3LYP 0.0 – – 1.58, 0.124 – 1.88, 0.031 2.04, 0.012
A0 and A1 are the C3NVH12and C19NVH28model clusters, respectively. The basis sets used were the following, in
A0: 6-3111G* for C and N atoms and 6-31G for H; in A1: 6-311G* for four central atoms, 6-31G* 1 s(0.0438) for atoms 5–16 in the second layer and 6-31G for all other atoms.
a
Davidson-corrected MRCI energies are given in italic.
b
bonds around the vacancy. The calculated CASSCF excitation energies are underestimated as the experimental vertical transition
3
A2-3E energy should be about 2.15 eV,25 but expansion of the
active space toward vacant orbitals leads to a better description of the excitation energies (Table 4). Apparently, in this system the inclusion of empty MOs is essential for more complete account of electron correlation.
The use of the MRCI method gives slightly better results, however, the transition energies remain too low. The second excited triplet state 23E is only 0.2–0.3 eV higher than 13E, but has much lower oscillator strength (Table 4). State-averaged cal-culations for two states, 13A2 and 13E, instead of three states,
13A
2, 13E, and 23E, generate slightly higher energies and there
is no substantial difference between the results for the small C3NVH12 and moderate C19NVH28 clusters. The lowest singlet
state 11E is found to lie 0.4–0.5 eV higher in energy than the ground state triplet and the second singlet 11A1 state is 0.1–
0.2 eV lower than the first excited triplet 13E, the experimental energy gap between them being about 0.04 eV. Although these calculations produce a qualitative correct picture of the lowest excited states, the quantitative description is flawed as the exci-tation energies are significantly underestimated. In our opinion, the main reason for these discrepancies is the use of a rather limited active space in the MRCI calculations. Additionally, we tried to test the CIS(D) approach64,65for this system. It produces reasonable energy difference between the 13A2 and 1
3
E states, but gives 13A1as the lowest excited triplet instead of 13E (Table
4). This matches neither experimental observation nor the results of multireference calculations, and so the single-determinant-based CIS(D) method is not appropriate here.
Now we consider higher excitations of the NV21center. Cal-culations were carried out using the moderate C19NVH28model
cluster with several basis sets and active spaces. So as only the Cs point group instead of C3v was utilized, the active spaces
necessarily included both symmetry-equivalent a0and a@ orbitals originated from e MOs. Nevertheless, at the CASSCF(8,15) level with active space including 8 electrons on 15 MOs, the energies
and oscillator strengths f of the A0 and A@ states degenerate within C3vsymmetry are slightly different. For instance, the
dif-ference in energies of the 13A0 and 23A@ components of 13E is 0.07 eV and f (23A@) is equal to approximately a half of the f
(13A0) value (see Table 5). When the active window is shifted up toward vacant orbitals and the two lowest MOs with occupa-tion numbers of 1.98 are excluded in the (4,14) active space, the splitting in the properties of the symmetry-equivalent states is much lower. Additionally, this leads to higher excitation ener-gies. With the B1 and B2 basis sets including diffuse functions on the vacancy-related atoms, the components of 13E state, 13A0, and 23A@, show moderate f values; the calculated oscillator strengths are higher for the components of 43E, 43A0, and 53A@,
whereas for the other states they are lower. The addition of dif-fuse functions for the atoms from the second layer around the vacancy (B1a and B2a basis sets) leads to a decrease of excita-tion energies for all considered states and the f values become highest for the 13E components, followed by those for the 53E state, while the other oscillator strengths are calculated to be low (Table 5). For singlet states calculated with the (4,14) active space and B2a basis set at the CASSCF level, the lowest 11E state is about 0.4 eV higher than the ground triplet state and the first excited singlet (11A0) is 0.01 lower then first excited tri-plet 13E (Table 5). Summarizing, there is no principal differen-ces in the results of CASSCF calculations for lowest excited states. At this level of theory, the qualitative picture seems to be described reasonably well, but the excitation energies are signifi-cantly underestimated.
Next, we considered the effect of dynamic electron correla-tion using CASPT2 calculacorrela-tions with the MOLCAS program package.63As before, the calculations were carried out with two
different basis sets (B3 and B3a, see Table 6) and two active states for CASSCF calculations, wave functions from which were used as initial approximation for CASPT2. The more com-plete account of electron correlation leads to an increase of the calculated excitation energies. With the best available basis set B3a and (4,16) active space including as many empty MOs as
Table 5. Relative Energies (in eV, With Respect to the 13A@ State) and Oscillator Strengths (in bold) for
10 Lowest States of the NV2Defect Calculated with the C19NVH28Model Cluster at the Ground
Triplet State Geometry.
State CAS(8,15) B1 CAS(4,14) B1 CAS(4,14) B1a CAS(4,14) B2 CAS(4,14) B2a State CAS(4,14) B2a 13A0 1.03, 0.067 1.39, 0.051 1.00, 0.154 1.36, 0.028 1.09, 0.158 11A@ 0.42 23A@ 1.10, 0.036 1.40, 0.053 1.00, 0.150 1.36, 0.028 1.10, 0.156 11A0 0.42 33A@ 1.14, 0.012 1.52, 0.001 1.23, 0.016 1.46, 0.000 1.31, 0.008 21A0 1.09, 0.349 23A0 1.15, 0.000 1.52, 0.005 1.23, 0.015 1.47, 0.000 1.31, 0.006 21A@ 1.26, 0.012 43A@ 1.19, 0.013 1.52, 0.001 1.35, 0.004 1.49, 0.018 1.43, 0.004 31A0 1.26, 0.008 33A0 1.20, 0.004 1.54, 0.000 1.33, 0.000 1.49, 0.017 1.40, 0.000 31A@ 1.37, 0.073 43A0 1.38, 0.083 1.74, 0.101 1.45, 0.035 1.69, 0.085 1.52, 0.028 41A@ 1.51, 0.115 53A@ 1.49, 0.068 1.74, 0.103 1.45, 0.034 1.69, 0.088 1.52, 0.028 41A0 1.52, 0.113 63A@ 1.67, 0.020 2.03, 0.020 1.53, 0.092 2.01, 0.025 1.60, 0.090 51A@ 1.91, 0.149 53A0 1.78, 0.025 2.16, 0.021 1.53, 0.089 2.11, 0.021 1.60, 0.087 51A0 1.92, 0.168
State-averaged CASSCF calculations with equal weights for A0and A@ states were employed. The basis set notation is the following: B1, 6-3111G* for N, C0, and C@ atoms, 6-31G* for C atoms 5–16 in the second layer, 6-31G for all other atoms; B1a – 6-3111G* for C0 and C@, 6-31G 1 s(0.0438) for atoms 5–16, 6-31G for all other
atoms; B2, 6-311G* for N, C0, C@, 6-31G* for atoms 5–16, 6-31G for all other atoms; B2a, 6-311G* for N, C0, C@,
possible, the calculated lowest excitation energy is underesti-mated by 0.35 eV as compared to the experimental value (Table 6). The energetic order of singlet states does not changed significantly after the refinement at the CASPT2 level in com-parison to that obtained at CASSCF.
As was discussed earlier, geometric parameters optimized for the moderate C19NVH28 model cluster are slightly different as
compared to those for the larger model systems, mainly due to the shift of the vacancy-related atoms away from the vacancy center (Table 2). The calculations with the geometric parameters opti-mized for the C49NV21 model cluster lead to a moderate increase
of about0.1 eV for the energies of the 13E–43E states and for the 53E state the energy change is more significant (Table 6). This reduces the deviation of the calculated lowest excitation energy for the 13E state from experiment to0.25 eV. For singlet states notable changes can be seen only for 11E and 41E.
In summary, combining the CASSCF and CASPT2 results, we predict the following sequence of electronic states for the NV2defect in diamond:
13A2, 11E, 11A1, 13E, 21E, 11A2, 23E, 31E, 33E, 43E, 41E, 53E
The significant probabilities for excitations from the ground state according to the computed oscillator strengths are expected for the 13E and 53E states; the second adsorption band should be 0.8–1 eV higher in energy as compared to the first one. For singlet states, the oscillator strength between 11E and 11A1 is
large (Table 5), and so one can expect fast 11A1? 11E
de-exci-tation. It follows from this result that there exist a possibility to overcome the shelving effect for this center by stimulating the 11E? 13A2conversion.
Excited States of the NV0Defect
Within the single-determinant approach, the components of the wave function of the doubly degenerate 12E ground state of the
model C19NVH28 cluster are constructed in the following way:
. . .42a0230a@2
43a0244a0245a0131a@0 (12A0) and 42a0230a@243a 02 44a0231a@145a00 (12A@) within C
s symmetry. Here, the 43a0 MO
(HOMO-1) corresponds to the nitrogen lone pair and 44a0 (HOMO), 31a@, and 45a0(SOMO) orbitals are composed from the broken bonds of the carbon atoms located around the vacancy. The 44a0 MO is doubly occupied and placing one electron on the degenerate 31a@ or 45a0 MOs gives the 12
A0or 12A@ components of the degenerate ground state. Hence, one can expect that the lowest excited states will be formed by transferring one electron from the doubly occupied HOMO-1 or HOMO to the empty or singly occupied 45a0or 31a@ SOMO.
B3LYP calculations for NV0 give the ground state wave function with nonequivalent occupations of the degenerate e orbitals and TD-B3LYP calculations with the 6-3111G* basis set at vacancy related atoms and 6-31G* at all others produce 4 lowest excited states in the range from 2 to 3 eV. In contrast to NV21, not only electrons from the broken bonds of three-coordinated C atoms around vacancy and diffuse functions but also the N lone pair participate in their formation. However, CASSCF calculations give a rather different qualitative result. Test computations with the minimal active space including only 44a0, 45a0, and 31a@ MOs generate the following wave functions for the ground state components: [email protected] and [email protected] with separate treatment of the A0 and A@ states, or [email protected] the case of state-averaged calcu-lations. The 22A0 and 22A@ excited states in this case have very high energies of 5–6 eV, and the wave functions for these states do not correspond to a simple electron transfer from 44a0to 45a0 or 31a@ because the composition of these MOs in excited states is significantly changed, with appearance of large contributions from diffuse functions. The active space extension by including the empty MOs, which ensured reasonable results for the nega-tive defect, does not change the qualitanega-tive picture for the neu-tral cluster; then the changes in orbital occupation numbers do
Table 6. CASPT2 Relative Energies (in eV, With Respect to the 13A@ state) for 10 Lowest Excited States of
the NV2Defect Calculated with the C19NVH28Model Cluster at the Ground Triplet State Geometry.
Triplet states
Active space and basis set
Singlet states
Active space and basis set
(8,15) B3 (8,14) B3a (4,16) B3a (4,16) B3aa (8,15) B3 (8,14) B3a (4,16) B3a (4,16) B3aa
23A@ 2.10 1.55 1.79 1.91 11A@ 0.15 20.06 0.34 0.63 13A0 2.10 1.57 1.79 1.91 11A0 0.28 0.15 0.35 0.62 33A@ 2.25 1.82 2.09 2.21 21A0 1.51 1.31 1.59 1.63 23A0 2.25 1.85 2.07 2.21 21A@ 2.14 1.65 1.97 1.96 43A@ 2.30 1.99 2.23 2.33 31A0 2.18 1.66 1.97 1.96 33A0 2.31 1.93 2.20 2.30 31A@ 2.20 1.83 2.15 2.19 43A0 2.48 2.05 2.30 2.36 41A@ 2.42 1.98 2.30 2.33 53A@ 2.53 2.05 2.31 2.36 41A0 2.41 1.99 2.29 2.31 63A@ 2.67 2.20 2.38 2.75 51A@ 2.62 2.12 2.47 2.70 53A0 2.66 2.18 2.39 2.74 51A0 2.60 2.22 2.48 2.65
State-specific calculations were carried out separately for A0 and A@ states. The basis set notation is the following: B3, aug-cc-pVDZ for N, C0, and C@ atoms, 6-31G* for C atoms 5–16, 6-31G for all other atoms; B3a, the same as B3, but with 6-31G*1 s(0.0438) for atoms 5–16. The experimental vertical3A2-3E excitation energy is2.15 eV.25
a
not exceed 0.01, but for the 22A0 and 22A@ states the excitation energies are lowered to 3.9 eV, and the 46a0 MO gets involved in the 22A0 wave function. With extended active spaces includ-ing several empty MOs, the lowest excitations correspond to the electron transfer not from 44a0, but from 45a0and 31a@ to higher empty MOs.
The extension of the active space down by inclusion of the doubly occupied 43a0 MO (5 active electrons), does not cover the nitrogen lone pair. Alternatively, according to the character of the computed CASSCF wave function, a lower MO corre-sponding to CC bonds of the vacancy-related atoms with their neighborhood gets into the active space, and the MO of the nitrogen lone pair remains out of the active space. Additionally, this leads to nonequivalent participation of the degenerate MOs in the active space. To restore the symmetry equivalence of these orbitals, one needs to additionally include the 30a@ orbital and then the active space becomes (7,15). For the same reason, the active space with nine active electrons violates the equiva-lence of degenerate MOs and the nitrogen lone pair is again pushed out of the active space. The relative energies of the low-est excited states calculated with all considered active spaces remain overestimated. A significant decrease of CASSCF excita-tion energies (by1 eV) can be achieved by adding highly dif-fuse functions (with exponent of 0.01) at atoms from the first and second layers (see Table 7, B4a basis set). Apparently, all the transitions involved correspond to electron transfer to diffuse Rydberg-type orbitals.
The inclusion of dynamic electron correlation at the CASPT2 level however gives a rather contradictory result. Transition energies calculated with the B3a basis set with moderate diffuse functions decrease by 0.5–0.6 eV to 3.3–3.5 eV (Table 7). This is not sufficient to attain the experimental value, which should be close to 2.4–2.5 eV for the vertical excitation, if the differen-ces between ZPL and the maximum of PA band for NV0 and NV2 are similar; note that ZPL for NV0 has been assigned to 2.16 eV.28,29In contrast, the CASPT2 excitation energies calcu-lated with the B4a basis set show an increase by up to 1 eV as compared to the CASSCF values (Table 7). The most probable reason of this failure is incorrect prediction of the order of excited states by CASSCF resulting in the absence of the
appro-priate states among available lowest CASSCF roots. To test this supposition we carried out calculations with the B3 basis set and (9,6) active space, in which we artificially limited participation of diffuse functions. Then, the lowest CASSCF excitation ener-gies are rather high, in the range of 4–6 eV, and the 43a0 MO (the N lone pair) is involved in the excitations in addition to the doubly occupied 44a0MO, which corresponds to a combination of broken bonds of C atoms around the vacancy. In this case both of these orbitals represent a mixture of the N lone pair and C broken bonds. When dynamic correlation is taken into account at the CASPT2 level, the excitation energies to these valence states drastically decrease; for instance, the value for 22A@ becomes2.4 eV (Table 7). The main contribution to the result-ing CASPT2 wave function of the 22A@ state comes from the 32A@ state at the CASSCF level (0.72), but the coefficients of
the 22A@, 42A@, and 52A@ states are also significant (0.36–0.42). When the active space is extended by including additional empty orbitals as in (9,7) and (9,8) spaces, total energies of both the ground 12E and 22A@ states increase because in this case the contributions of higher diffuse states in the wave function rise, while the weights of the lower states reduce. As a result, the 12E ? 22A@ excitation energy does not change significantly. The energy variations for the 22A0and 32A@ states are more sub-stantial; 22A0is lower than 32A@ with the (9,6) active space, but
the order is reversed with the (9,7) and (9,8) active spaces (Ta-ble 7). So as the energy difference between 22A0 and 32A@ is not large, it is quite likely that these states are components of 22E, but the size of the active spaces used is not sufficient to achieve proper degeneration.
Combining the results obtained at the CASPT2 level with the B3 and B4a basis sets we predict the following order of elec-tronic states for the NV0defect: 12E (0.0), 12A2(2.4 eV), 22E
(2.7–2.8 eV), 12A1, 3 2
E, and so on (3.2 eV and higher). One can see that the agreement of the CASPT2 calculated lowest ex-citation energy with experiment28,29is quite reasonable. The first group of excitations involves electron transitions from HOMO-1 and HOMO to SOMO with participation of broken bonds of threefold-coordinated C atoms and the nitrogen lone pair, i.e., these are valence excited states. The second group corresponds to electron transitions from SOMO to highly diffuse states,
simi-Table 7. CASSCF and CASPT2 (in Bold) Relative Energies (in eV, With Respect to the 12A0State) for
Lowest Excited States of the NV0Defect Calculated with the C
19NVH28Model Cluster at the Ground
Doublet State Geometry.
Active space 12A@ 22A@ 22A0 32A@ 32A0 42A@ 42A0 (3,15), B3a 0.01, 0.00 4.14, 3.63 4.05, 3.47 4.90, 4.28 4.18, 3.98 5.03, 5.03 4.58, 4.26 (5,16), B3a 0.23, 0.04 3.89, 3.32 4.04, 3.28 4.29, 4.61 4.17, 4.02 4.55, 4.94 4.57, 4.28 (7,15), B3a 0.03, 0.07 3.89, 2.90 4.10, 3.62 4.67, 3.11 4.30, 3.95 4.70, 4.64 5.04, 4.24 (9,15), B3a 0.94, 0.00 4.79, 3.29 4.02, 3.32 5.11, 4.52 4.11, 3.98 5.45, 5.28 4.53, 4.83 (9,13), B4a 20.64, 0.20 2.44, 3.49 2.89, 3.23 2.77, 3.83 3.08, 3.48 2.80, 3.86 3.14, 3.51 (9,6), B3 0.08, 0.06 3.83, 2.38 4.36, 2.64 4.38, 2.88 6.01, 3.47 6.15, 4.38 6.26, 4.39 (9,7), B3 0.08,20.16 4.33, 2.36 4.65, 2.85 4.72, 2.60 5.34, 3.93 6.21, 4.36 5.86, 4.64 (9,8), B3 20.09,20.07 4.07, 2.29 4.43, 2.75 4.44, 2.60 5.10, 3.97 5.92, 4.33 5.67, 4.54
State-specific calculations were carried out separately for A0 and A@ states. The basis set notation is the following: B3, aug-cc-pVDZ for N, C0, and C@ atoms, 6-31G* for C atoms 5–16, 6-31G for all other atoms; B3a, B3 1 s(0.0438) for atoms 5–16; B4a, B31 sp(0.01) for vacancy-related atoms 1 s(0.01) for atoms 5–16.
lar to Rydberg states for molecules. To some extent, the first excited 12A2 state in the NV0 system is similar to the 13A2
ground state in NV2, but with the hole delocalized at HOMO-1 and HOMO and involving symmetrical combination of carbon broken bonds and the nitrogen lone pair. According to the com-puted Mulliken populations the excitation to 12A2 leads to a
transfer of0.5 e from N to the vacancy-related C atoms. The oscillator strength for the 12A
2 ? 12E de-excitation evaluated
using the transition dipole moment calculated by projecting CASSCF contributions onto the CASPT2 wave function is 0.02.
Concluding Remarks
We have performed quantum chemical modeling of the nitro-gen-vacancy point defect in diamond employing a variety of the-oretical methods and using finite NCnHm clusters with hydrogen
atoms saturating broken CC bonds on the boundary. The HF, B3LYP, and CASSCF approaches produced similar results for the geometry and electronic structure of the ground states for model clusters simulating the NV0and NV2defects. In terms of geometry changes introduced by the defect, the atoms located directly near the vacancy are shifted from the vacancy center by 0.10–0.12 A˚ , whereas the positions of the atoms in the next layer remain nearly unchanged indicating a local character of geometry relaxation due to the defect. It is found that unpaired electrons in the doublet and quartet states of the neutral defect and the triplet ground state of the negatively charged defect are localized mainly on the three carbon atoms around the vacancy positioned in the center of model clusters and the electronic den-sity on the N impurity atom and C atoms bound to it is weakly disturbed as compared to defect-less clusters. These results indi-cate that the nitrogen-related electronic distribution is rather rigid and is weakly spin polarized. The effect of the cluster size on the defect properties appears to be small; reasonably accurate results can be achieved with clusters containing only one addi-tional carbon layer around the defect.
Wave functions of the lowest excited states of NV0and NV2 show a strong multireference character and contain significant contributions from diffuse functions. According to all calcula-tions at the TD-DFT, CASSCF, MRCI, and CASPT2 levels, the lowest excited states involve various distributions of electrons on molecular orbitals localized close to the cluster center, i.e., the vacancy position, and originated from the broken bonds of threefold coordinated C atoms surrounding the vacancy and the N lone pair for NV0system. The contributions from the MOs re-sponsible for the bonding of the vacancy-related atoms with the rest of the bulk are insignificant.
CASSCF calculations give underestimated excitation energies for the negative defect and overestimated energies for the neu-tral system. The inclusion of dynamic electronic correlation effects at the CASPT2 level leads to a reasonable agreement (within 0.25 eV) of the calculated transition energy to the lowest excited state with the experimental value for the NV2 system. Several excited states have been found in the energy range from 2 to 3 eV, but, according to the calculated oscillator strengths, only for the 13E and 53E states the excitation probabilities from
the ground state are expected to be significant. Here, the second adsorption band is predicted to lie 0.8–1 eV higher in energy as compared to the first one. For singlet states, a large oscillator strength was computed between 11E and 11A1, and one can
expect the 11A1 ? 11E de-excitation process to be fast.
There-fore, it may be possible to overcome the shelving effect for this defect center by stimulating the 11E? 13A2conversion.
For the neutral NV0 defect center, CASPT2 calculations
result in significant reduction of transition energies for excited states involving valence orbitals as compared to the CASSCF values; for electronic states with high contributions of diffuse orbitals the difference between CASPT2 and CASSCF is small. As a consequence, the energetic order of excited states at the CASPT2 and CASSCF levels is different and the CASPT2 cal-culated excitation energy for the first excited 12A2 state is in
reasonable agreement with the experimental value. Both for the NV2 and NV0 systems, taking into account dynamic electron correlation at the CASPT2 level leads to the most substantial energy decrease for the 1A2 state. In the negatively charged
defect this is the ground state and the excitation energy increases, whereas in the neutral cluster this state is excited and hence the excitation energy is reduced. Nevertheless, for both defects the CASPT2 method appears to be adequate to describe the transition energies with reasonable accuracy.
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