DFT Calculation of RussellâSaunders Splitting for Lanthanide Ions

Article pubs.acs.org/JPCC

DFT Calculation of Russell−Saunders Splitting for Lanthanide Ions Doped in Hexagonal (β)-NaYF4 Nanocrystals Ge Yao, Mary T. Berry, P. Stanley May, and Dmitri Kilin* The University of South Dakota, Vermillion, South Dakota 57069, United States S Supporting Information *

ABSTRACT: A systematic investigation is reported of the optimized geometry and electronic structure of trivalent lanthanide ions (Ln 3+) doped in hexagonal (β)-NaYF4 nanocrystals in the basis of density functional theory with a spin polarization approach. A model Na24Y23Ln1F96 nanocrystal with a single central lanthanide dopant (Ln3+) is used. Electron spins couple to give a total spin, S, and electron orbital angular momenta couple to give total orbital angular momentum, L. Spin−orbit coupling is neglected in this initial study. Several key observables are found to be strongly related to the number of unpaired f-electrons in the model. After geometry optimization, the phenomenon of the lanthanide contraction is observed, and the configurations of 4f-electron-like orbitals satisfy Hund’s Rule under the orbital decomposition. Spinpolarized density functional theory is applied to generate the Russell−Saunders terms (2S+1L) terms of lanthanide ions. The energy differences between the first and the second terms are calculated and show good agreement with experimental measurements. The free-ion-like behavior of Ln3+ ions in the nanocrystal is observed as well.

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quantum number, and 2S+1 is the total spin multiplicity.5 Thus, the lanthanides tend to exhibit rich and complex energy-level structures.6 In the context of this project, it is useful to consider the splittings within a particular [Xe]4fn configuration as arising from the different interactions in the individual microstates, where each microstate is defined by a specific distribution of electrons within the 4f orbitals, including the spin states within those orbitals. The individual microstates will exhibit different levels of electron−electron repulsion, spin−orbit coupling, and crystal-field interactions. Various density functional-theory (DFT)-based computational techniques7−9 have been developed for the ab initio calculation for the properties of lanthanide compounds within a projected augmented wave basis in spin-polarized,10,11 spin−orbit coupling,12,13 Hubbard U corrected local-density approximation (LDA + U),14 Armiento and Mattsson functional,15,16 and dynamically screened (GW) approaches.17,18 A method that could describe these interactions in a computationally inexpensive manner would be valuable for understanding lanthanide spectroscopy. One excellent strategy to accomplish this is through constrained density functional theory (C-DFT), which is a combination of standard DFT techniques with additional conditions, called “constraints”.19 These conditions limit the possible electronic configurations, among which DFT searches for a ground state. As an example, one can perform a search

INTRODUCTION Computational studies of trivalent lanthanide (Ln3+) compounds are considered challenging due to the complexity of the [Xe]4fn electronic structure.1,2 4f-in-core relativistic effective core potentials (RECPs) are normally applied as an approximation in lanthanide calculations, since 4f electrons do not significantly contribute to bonding compared to the 5d and 6s orbitals.1,3 However, this strategy is not appropriate when the spectroscopic properties of lanthanide ions are under consideration. As modeled in this work, Ln3+ ions act as optically active centers when embedded in a transparent host material. Doping is typically done at low levels and results in substitution at host-ion sites. The dopant ions exhibit luminescence in the UV, visible, or near-infrared spectral regions under an appropriate excitation via 4f → 4f transitions in lanthanide ions.4 Therefore, 4f electrons must be considered explicitly as valence electrons to accurately describe their optical properties. The nominal [Xe]4fn electronic configurations of lanthanides exhibit splitting in energy due to a variety of interactions. The Coulombic interaction from electron−electron repulsion between 4f electrons creates the largest splitting of energies, with separations on the order of 104 cm−1. Spin−orbit coupling creates splitting on the order of 102−103 cm−1 and interaction with the lattice, usually referred to as the crystal-field effect, creates splitting on the order of 10−102 cm−1. The Coulombic interaction, prior to turning on spin−orbit coupling, results in multiple levels described by the Russell−Saunders term symbols, 2S+1L, where L is the total orbital angular momentum quantum number, S is the total spin angular momentum © 2013 American Chemical Society

Received: April 28, 2013 Revised: July 16, 2013 Published: August 14, 2013 17177

dx.doi.org/10.1021/jp404206e | J. Phys. Chem. C 2013, 117, 17177−17185

The Journal of Physical Chemistry C

Article

among only those electronic states which provide desired values of the total magnetic moment of the system.20,20b Previously, the crystalline structure of hexagonal-phase (β)NaYF4, a promising host material for fluorescent-phosphors,21−25 was successfully modeled with different sizes, number of ions, volumes, and shape.26 In the current study, a series of lanthanide ions, La3+, Ce3+, Pr3+, Nd3+, Pm3+, Sm3+, Eu3+, and Gd3+, which have [Xe]4fn electron configurations with n ranging from 0 (unoccupied f orbital) to 7 (halfoccupied f orbitals) doped in β-NaYF4 are modeled to investigate the energy differences between their Russell− Saunders terms by applying constrained DFT in a systematic manner. The spin polarization approach in describing the electronic interaction is used as the first step to study the energy levels of the lanthanide ions. The subenergy splitting by magnetic interactions (spin−orbit coupling) of the order of 102 −103 cm−1 is neglected in the calculation and left for future consideration.

υβ [⇀ r , ρβ (⇀ r )] =

β

iβ

number f iα or f iβ, which take on values of 0 or 1. ρα (⇀ r)=

ρβ (⇀ r)=

Nα = Nβ =

(3)

(10)

ΔETOT(spin−flip) = ETOT(Nα − 1, Nβ + 1) (4)

− ETOT(Nα , Nβ) ΔE ORB(spin−flip) = εβ ,LUMO(β) − εα ,HOMO(α)

(11) (12)

2S+1

A Russell−Saunders term L is assigned in 4f electron energy splitting calculation, where the total angular momentum quantum number (L) value is obtained as the largest value among the all possible angular momentum projections, the total spin quantum number (S) is simply the sum of the ms = 1/2 of the unpaired f-electrons. For comparison, the spectroscopic investigations of lanthanide doped LaF3 reported by Carnall et al.32 is applied to calculate the energy difference between the first and second terms. The energy for each term E(2S+1L) is calculated as the weighted average of the energies of the reported 2S+1LJ states, with the weighting equal to 2J + 1, the degeneracy of the 2S+1LJ state.

iβ

eigenenergies. The total energy functional ETOT [ρα,ρβ] does r ), and exist for given electron densities ρα (⇀ r ) and ρβ (⇀ includes three interactions: Coulomb, exchange, and correlation. The potential can be expressed as a functional derivative of the exact energy functional ETOT [ρα,ρβ] − T with respect to variation of the spin component density (eq 5) and (eq 6), where T represents kinetic energy in eq 3 and 4.

α

∫ ρβ dr = ∑ fiβ

The sum of (Nα + Nβ) is fixed and equals the total number of electrons. The difference (Nα − Nβ) is often referred to as the spin polarization parameter, and is calculated as an output of the DFT procedure. Alternatively, one can fix this value (Nα − Nβ = constant), which establishes a constrained DFT calculation in constructing the ground or excited terms. To calculate the energy of the first excited term, a 4f electron is flipped from the HOMO(α) to LUMO(β) and the optimized orbitals and relevant energies are recalculated. The energy difference between the ground and first excited term may then be calculated as in eq 11. An approximation to this energy difference may also be calculated from the energies of HOMO(α) and LUMO(β) orbitals directly (eq 12) and is included for comparison as well.

where the first and second terms in each equation correspond r) to kinetic energy (Tα,β) and potential energy (να,β), φiαKS(⇀ KS ⇀ and φ ( r ) are the KS spin orbitals, and εiα and εiβ are

δ(E TOT[ρα , ρβ ] − Tα) δρ (⇀ r)

(8)

(9)

i

Then, the Kohn−Sham (KS) equation in spin polarization can be expressed as

υα[⇀ r , ρα (⇀ r )] =

∫ ρα dr = ∑ fiα i

(2)

⎛ ℏ2 2 ⎞ r , ρβ (⇀ r )]⎟φiβKS(⇀ r ) = εiβ φiβKS(⇀ r) ∇ + υβ [⇀ ⎜− ⎝ 2m ⎠

(7)

Equations 3−8 are solved in an iterative manner in VASP. The orbital character is analyzed by projecting the wave functions onto spherical harmonics within spheres of a Wigner−Seitz radius around each ion.30 The output contains the decomposition of hybrid Kohn−Sham orbitals into the form of s, p d, and f orbital contributions from each ion to each hybrid Kohn− Sham orbital. It is important to note that the total number of electrons can be different for spin α and β, as represented in the following equations, which is the basic concept for constrained DFT.31

ρα − ρβ

⎞ ⎛ ℏ2 2 r , ρα (⇀ r )]⎟φiαKS(⇀ r ) = εiαφiαKS(⇀ r) ∇ + υα[⇀ ⎜− ⎠ ⎝ 2m

r )|2 ∑ fiβ |φiβKS(⇀ i

(1)

ρ

r )|2 ∑ fiα |φiαKS(⇀ i

COMPUTATIONAL DETAILS Density functional theory with the generalized gradient approximation (GGA)27 implemented in the Vienna ab initio simulation package (VASP)28 is employed for the structural optimization and total energy calculations. Total energy r) ETOT [ρα,ρβ] is expressed in terms of ρα (⇀ r ) and ρβ (⇀ which are the spin-up and spin-down electron densities, corresponding to the diagonal elements in spin space, and r ), does which play the same role as total electron density, ρ(⇀ r) in the absence of a magnetic field.29 The electron density ρ(⇀ r ) of the Kohn−Sham (KS) orbitals for a and spin density ζ(⇀ given spin component can be expressed as in eqs 1 and 2, respectively.

ζ(⇀ r)=

(6)

The total density of electrons is the combination of partial r )|2 or charge densities of KS orbitals with known spin |φiαKS(⇀ |φ KS(⇀ r )|2 . Each of these contributions is taken with occupation

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ρ(⇀ r ) = ρα + ρβ

δ(E TOT[ρα , ρβ ] − Tβ) δρ (⇀ r)

(5) 17178



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Figure 1. Optimized structure (a,b) of Na24Y23Eu1F96 from different aspect views. Elements are color coded as blue = Na, gray = Y, yellow = Eu, and orange = F. (c) Average Ln−F bond distances (Å) of Na24Y23Ln1F96, Ln = La3+, Ce3+, Pr3+, Nd3+, Pm3+, Sm3+, Eu3+, Gd3+, Y3+. (d) Schema of europium ion coordination.

E(2S + 1L) =

∑ (2J + 1) E(2S+ 1LJ)/∑ (2J + 1) J

J

calculate the Kohn−Sham (KS) ground-state until the difference of the total energy is less than 0.0001 eV.

(13)

■

The density of states, the number of states per interval of energy available to be occupied by electrons, is defined separately for spin α and β by sums of the Dirac delta function. Dα (ε) =

∑ δ(ε − εiα) i

Dβ (ε) =

∑ δ(ε − εiβ) i

RESULTS AND DISCUSSION

A. Geometry Optimization and Structural Properties. Based on our previous model of NaYF4,26 we chose βNa24Y24F96 as the host material in which to dope single lanthanide ions as Na24Y23Ln1F96 (Ln = La, Ce, Pr, Nd, Pm, Sm, Eu, Gd). The optimized molecular structure of Na24Y23Ln1F96 is shown in Figure 1, using Na24Y23Eu1F96 as an example. The lanthanide contraction is a phenomenon of the ionic radius decreasing with the increase in atomic number. It is exhibited while filling the inner 4f electron shell across the lanthanide series. From La3+ (radius 1.16 Å) to Gd3+ (radius 1.05 Å), the radius decreases by 9%.34 In our simulation, the lanthanide contraction is also observed, with the Ln-F bond length decreasing by 3.3% from La3+ to Gd3+, consistent with a cationic radius decrease of 6.6%. The smaller effect of the lanthanide contraction observed in our simulation may be

(14)

(15)

Here, Dirac delta function is approximated by a Gaussian profile with finite value of the width parameter, 0.05 eV, as in refs 33(a)−33(b). The Perdew−Burke−Ernzerhof (PBE)33c functional is chosen in calculation. The k-meshes are generated in the Brillouin zone with a 1 × 1 × 1 Monkhorst−Pack grid in each direction. VASP uses a self-consistency cycle with a Pulay mixer and an iterative matrix diagonalization scheme to 17179



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Table 1. Calculated Ln−F Bond Distances (Å) for the Nine Nearest Neighbor Fluoride Ions in the Model Na24Y23Ln1F96 Nanocrystal with Ln = La3+, Ce3+, Pr3+, Nd3+, Pm3+, Sm3+, Eu3+, Gd3+ La3+−F− Ce3+−F− Pr3+−F− Nd3+−F− Pm3+−F− Sm3+−F− Eu3+−F− Gd3+−F− Y3+−F−

1

2

3

4

5

6

7

8

9

avg

2.218 2.207 2.19 2.185 2.168 2.171 2.14 2.147 2.119

2.222 2.209 2.185 2.177 2.166 2.152 2.167 2.166 2.189

2.257 2.248 2.234 2.217 2.211 2.173 2.185 2.194 2.214

2.318 2.306 2.286 2.27 2.263 2.233 2.242 2.248 2.252

2.323 2.305 2.275 2.268 2.26 2.244 2.254 2.231 2.249

2.339 2.323 2.307 2.309 2.286 2.302 2.261 2.255 2.253

2.383 2.364 2.35 2.351 2.331 2.327 2.32 2.317 2.269

2.387 2.371 2.354 2.342 2.332 2.339 2.324 2.291 2.258

2.499 2.482 2.463 2.449 2.436 2.435 2.435 2.4 2.271

2.327 2.313 2.294 2.285 2.273 2.264 2.259 2.25 2.23

Table 2. Number of Electrons of Na24Y23Ln1F96 Occupying Spin α and Spin β Orbitals after Spin Polarization total (Nα + Nβ) spin α (Nα) spin β (Nβ) no. of unpaired e− (Nα − Nβ) magnetization

La

Ce

Pr

Nd

Pm

Sm

Eu

Gd

960 480 480 0 −0.01470

961 481 480 1 1.00214

962 482 480 2 2.00057

963 483 480 3 2.99974

964 484 480 4 3.99867

965 485 480 5 5.00004

966 486 480 6 6.00002

967 487 480 7 6.99645

explained by the dominance of Y3+ in determining the crystal structure, where the lanthanide ions are impurities. Each lanthanide is bound to nine nearest-neighbor fluoride ions with bond distances ranging between 2.1 and 2.5 Å. For each lanthanide, the nine bond distances are listed in ascending order in Table 1. The average value is used to illustrate the lanthanide contraction in Figure 1(c, d). B. Orbital Distribution and 4f Electron Configuration. Pseudopotentials are used in VASP to save computational time, by which only valence electrons are treated explicitly. The electronic configuration of ions in this calculation can be given as as F = [He]2s22p7, Na = [Ne]3s1, Y = [Kr3d10]4s24p64d15s2, La = [Xe]5s25p65d16s2, Ce = [Xe]4f15s25p65d16s2, and Pr−Gd = [Xe]4fn5s25p66s2, (n = 3, 4, ...8). The summation of the number of electrons outside of the noble gas cores gives the total number of valence electrons in the calculation. Therefore, the total number of valence electrons for Na24Y23Ln1F96 can be calculated as 960 + n, where n is the number of 4f electrons on the lanthanide ion. In Table 2, the 4f electrons occupy one spin component, which is a result of Hund’s Rule One which describes how the orbitals are each occupied singly with electrons of parallel spin before double occupation occurs. The calculated magnetizations of the Na24Y23Ln1F96 unit cell from the response of the unbalanced magnetic dipole moments generated by the spin of the electrons are consistent with the number of unpaired electrons. The projected density of states (pDOS) for s, p, d, and f orbitals in Na24Y23Eu1F96 is shown as an example in Figure 2a. The DOS projections on s, p, and d orbitals are symmetric; in other words, the energy for partial (l = s, p, d orbitals) density states in spin (α) and spin (β) components are equal, with pDOSl,α(ε) ≈ pDOSl,β(ε), since those orbitals are closed shell with spin-paired electrons. In contrast, the DOS projections on f orbitals are drastically different for spin α and spin β components, since they are only partially occupied. This asymmetry is another manifestation of Hund’s Rule. In Figure 2b, the charge density of each 4f-like electron orbital in Na24Y23Eu1F96 for both spin (α) and spin (β) components are shown. Charge density is localized around the Eu3+ ion. Each orbital is a linear combination of several 4f-atomic orbitals. The

Figure 2. Case study analysis of electronic structure of Na24Y23Eu1F96. (a) Projected density of states of s (purple short dots), p (black short dashes), d (green dash), f (red solid), and total (blue dots) orbitals in spin-up (α) and spin-down (β) component in Na24Y23Eu1F96. (b) Partial charge density of 4f-electron-like orbitals in Na24Y23Eu1F96; upper row is spin (α) and lower row is spin (β) components. All images correspond to the isosurface value 0.03. Asterisk (∗) signs the occupied orbital. (c) Na24Y23Eu1F96 atomic orbital decomposition in KS orbital number for a broad range of orbitals.

atomic basis decomposition helps us to quantitatively analyze possible transitions between two orbitals of interest. All lanthanides in this research exhibit a similar s, p, d, and f-like 17180



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orbital distribution except Ce3+, which has strong mixing of d and f orbitals. In Figure 2c, decomposition of the KS orbitals into s, p f, and d projections are illustrated as a function of orbital number. A total seven of Eu3+ 4f dominated orbitals are observed for both spin (α) and spin (β), and the six 4f electrons reside in one spin direction, spin (α) only. Unlike other lanthanide ions discussed in the paper, the broad crystal-field splitting of the 5d level of Ce3+ by the host lattice may overlap the 4f levels.35,36 Therefore, it is not surprising to see the mixing of 4f and 5d like orbitals in Na24Y23Ce1F96. Unoccupied orbitals at the bottom of the CB, around 7 eV, are attributed to f-orbitals of Ce3+ and form a cluster of closely spaced peaks (Figure 3).

Rule that half or fully occupied configurations have lower relative energies. D. Density of States: First Term versus Second Term. La3+ has the noble gas configuration ([Xe]4f0). Therefore, no 4f → 4f optical transitions occur from the ground state. Similarly as for NaYF4 (Y3+ ion [Kr]4d0), NaLaY4 is a transparent host material, although the decrease in cation size of Y3+ versus La3+ is thought to cause an increase in the crystal-field strength around the dopant ions and to lead to enhanced UC efficiency.37 A comparison of the DOS for Na24Y23La1F96 and Na 24 Y 24 F 96 is shown in Figure 4. They are almost indistinguishable. As VASP does not support calculation for the swap of majority and minority spin components, Ce3+ is not calculated in this work. The lanthanide ions, from Pr3+ to Gd3+, display abundant 4f → 4f transitions. The lanthanide ion can be promoted to any energetically available state within the [Xe]4fn configuration under an appropriate excitation. Emission, however, often occurs from the spectroscopic level (2S+1LJ) included in the first and second Russell−Saunders terms due to rapid internal conversion.38 Therefore, the focus here is on the first and second terms calculation. A qualitative trend is observed in the computed 4f-orbitals. The spin majority (α) orbitals with partially or fully filled occupation have lower energy than the unoccupied orbitals for the spin minority component (β). While more and more 4f-orbitals are filled from Pr3+ to Gd3+, the energy of a filled orbital also experiences decrement of its value. The energy difference between spin-up and spin-down manifolds also grows from Pr3+ to Gd3+ which is consistent with the trend for an increase in the energy difference between the first and second Russell−Saunders terms (Figure 5). The spin (α) and spin (β) components of DOS show that both the energy and occupation of an orbital are spin direction dependent, εiα ≠ εiβ and f iα ≠ f iβ. This unsymmetrical character for the ions with partially filled 4-shells is consistent with Hund’s Rule. In calculating the first and second terms, the lowest energy level of crystal-field multiplet is chosen as an approximation to the term energy. There are two approaches to calculate the energy difference between the first and second terms in spin polarization. One is directly calculated by the energy difference for the relevant spin polarized orbitals as created in the ground state calculation (ΔEORB in eq 12); another is by flipping a 4f electron to the LUMO orbital of another spin configuration, recalculating the orbital energies under constrained DFT and taking the difference of the total energies (ΔETOT in eq 11). A systematic investigation of the energy levels of lanthanide ions in β-NaYF4 has not been published. Since LaF3 and βNaYF4 are fluorides that have similar phonon energies,39 and “YF3” structures of both are based on a distorted tricapped trigonal prismatic 9 coordination,40 the spectroscopic investigations of lanthanide doped LaF3 reported by Carnall et al.32 are used to compare to the calculated energy difference. This

Figure 3. Projected density of states of s (purple short dots), p (black short dashes), d (green dash), f (red solid), and total (blue dots) orbitals in spin (α) and spin(β) component in Na24Y23Ce1F96.

C. Differences in Total Energies between NonspinPolarized and Spin Polarization. In order to quantitatively analyze the influence caused by spin coupling in the electronic structure calculation, the DFT total energy was calculated in non-spin-polarized approach with total density only, and with spin polarization, based on two components of charge density, ρα and ρβ, as shown in Table 3. For La3+, with 4f0 in f electron configuration, the total energies do not change under either spinless or spin polarization conditions, since all electrons are paired. As the number of unpaired f electrons increases, the differences in the total energies between spinless and spinpolarized calculations increases, which is evidence of Hund’s Rule that for a given electron configuration, the term with maximum multiplicity has the lowest energy. Meanwhile, from Sm3+ to Gd3+, a decrease in total energy after spin polarization is observed. In particular, Gd3+, with a 4f7 configuration exhibits a significant decrease in total energy, which reflects Hund’s

Table 3. Comparison of the Total Ground-State Energy before and after Spin Polarization of Na24Y23Ln1F96 (eV)a

a

Ln

La

Ce

Pr

Nd

Pm

Sm

Eu

Gd

(Nα − Nβ) spin off spin on ΔE, eV

0 −876.113 −876.113 0.000

1 −876.487 −876.524 −0.037

2 −877.820 −877.621 0.198

3 −878.583 −878.731 −0.148

4 −880.263 −880.165 0.098

5 −881.077 −882.131 −1.054

6 −878.269 −880.184 −1.915

7 −882.917 −886.182 −3.265

The absolute value of the difference increases with the number of unpaired electrons (Nα − Nβ). 17181



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Figure 4. DOS of Na24Y23La1F96 (left) and Na24Y24F96 (right).

Figure 5. DOS of ground state(left) and first excited state (right) of Pr3+, Nd3+, Pm3+, Sm3+, Eu3+, and Gd3+. Each DOS panel is representing a term, which is labeled according to Russell −Sounders notation 2S+1L. Panels are arranged according to the increase of number of unpaired electrons. The procedure of assigning a term to computed electronic configuration is explained in main text. Interestingly, upon spin−flip constraint, the energies of orbitals with modified occupation numbers experience change in energy.

positions of the 2S+1LJ states, where the weighting is equal to 2J

data provides the positions of the spin−orbit coupled terms 2S+1 LJ. The position of the imagined 2S+1L state, from which these arose, is calculated as the weighted average of the

+1, the degeneracy of the 17182

2S+1

LJ state. The calculation for



Article

Eu3+7:F → 5D is illustrated with an energy level diagram as an example (Figure 6).

europium ion shows abnormal deviation from the trend in the energetic comparison in Figure 7. (If the data for Eu3+ is removed from the experimental trend line fitting, the slope of the new line becomes 5041 cm−1/electron (R2 = 0.9768), and the differences are by 2.3% and 8.1%, comparing to those of ΔETOT and ΔEEXP.) Two filled shells (5s, 5p) outside the 4f orbital shield the lanthanide ion from its environment.42 Therefore, the crystalfield effects in lanthanides ions are relatively weak, and lanthanide ions in compounds behave more like free ions than do transition metals.4 The orbital energy differences of lanthanide ions without the NaYF4 lattice were calculated by the same method as with the NaYF4 lattice. The differences between the free-ion and NaYF4 energy splitting (as determined by orbital energy difference calculations) are 1.7% for Pr3+, 2.5% for Nd3+, 0.8% for Pm3+, 11.6 for Sm3+, 0.8% for Eu3+, and 4.4% for Gd3+. The slope of the free ion orbital energy differences in number of 4f-electron dependence is 5640 cm−1/electron, which is by 3.5% only, compared to that of doped ions (Table 4 and Figure 7). The results agree with the hypothesis that the crystal field gives small perturbation to the lanthanide ions.

Figure 6. Calculation of energy difference between ground and first excited term (Eu3+ as an example). Each term consists of several states. Each state corresponds to a specific value of total momentum, J. The states with larger J have higher degeneracy and are weighed accordingly relative in calculating the average energy of the 2S+1L term. Energies of individual states are available from the report of Carnall et al.32

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SUMMARY Studies of the energy-level structure of lanthanide-doped complexes within the 4fn configuration are still at the limit of computational feasibility. Based on previous work on the hexagonal-NaYF4 matrix, we report successful modeling of the lanthanide-ion doped β-NaYF4 within the spin polarization regime. Geometry optimization across the Ln3+ series correctly predicts the phenomenon of the lanthanide contraction. The computed ground term electronic configuration is carefully analyzed using the projection of KS orbitals onto atomic orbitals, while the population of the orbitals with unpaired electrons follows Hunds’ Rule. The importance of the spinpolarized approach is verified by comparison of the total energy for spinless and spin polarized approaches. Here, the total energy increases when the spin polarization increases, especially for the half-filled electron configuration of Gd3+. The second term of each lanthanide ion is generated using constrained density functional theory. The energy levels in the Russell− Saunders terms (2S+1L) are calculated by two different methods: (i) total energy difference and (ii) orbital energy difference. The results obtained by calculating the total energy between the first and second term are compared with the experimental measurements. A linear increase of energy difference with the number of unpaired electrons is observed, except for the europium ion, in which spin−orbit coupling is comparable to the electrostatic interaction. The orbital energy differences of free lanthanide ions are calculated using the same method; the similarities between the lanthanide ions with and without the surrounding matrix prove the postulation that lanthanide ions have free-ion character even when residing in a chemical matrix. The computational modeling shows that several key properties directly depend on the number of unpaired electrons in lanthanide ions: specifically interatomic distance, spin polarization energy, and first excitation energy. The present study shows that it is possible to predict the 4f transition energies using the eigenvalues of 4f- and 5d-like orbitals calculated by a first-principles orbital structure method in spin polarization, and forms a basis for future inclusion of spin−orbit coupling interactions.

The simulation results both for total energy (ΔETOT, eq 11) and for orbital energy (ΔEORB, eq 12) difference have the same trend as the experimental results (ΔEEXP), as shown in Table 4 Table 4. Comparison of Calculation Results by Different Methods for Lanthanide as Doping or Free Ions to Measurements in Literature ΔEORB

ΔETOT

expt32

ΔEORB of free ion

Pr3+3H → 1G (cm−1) Nd3+4I → 2H (cm−1) Pm3+5I → 3K (cm−1) Sm3+6H → 4G (cm−1) Eu3+7F → 5D (cm−1) Gd3+8S → 6P (cm−1)

9464 16 006 21 095 26 172 29 012 38 778

3817 7191 12 562 16 500 24 098 27 366

7411 11 060 14 646 19 487 20 879 32 687

9627 15 601 20 934 23 146 29 252 40 472

slope (cm−1/electron) R2 slope difference from experiment

5448 0.9801 18.7%

4925 0.9886 7.3%

4591 0.9286

5640 0.9563

transition

and Figure 7. Interestingly, for each ion, ΔETOT < ΔEEXP < ΔEORB is observed. ΔETOT and ΔEORB grow linearly with number of unpaired electrons. The slopes of these linear functions are found as 4925 and 5448 cm−1/electron respectively. These differ by 7.3% and 18.7%, respectively, from the experimental value of 4591 cm−1/electron. The calculation based on the total energy difference shows the better fit to the experimental results. In general, the electrostatic repulsion splitting is an order of magnitude higher than spin− orbit coupling splitting. However, in the europium ion, the spin−orbit interaction is comparable to the coulumic interaction due to the Wybourne-Downer mechanism, in which the involved spin−orbit interaction among states of the excited configurations leads to an additional mixing of spin states into the 4f configuration contributing significantly in the construction of terms.41,41b This might be the reason that the 17183



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Figure 7. Energy of first excitation as function of number of unoccupied electrons. The slope of the fitted line for orbital energy differences (eq 12) is 5448 cm−1/electron; the slope of the fitting line for experimental observations, reported by Carnall et al.32 is 4591 cm−1/electron; the slope of the fitted line for total energy difference (eq 11) is 4925 cm−1/electron; and the slope of the fitted line for orbital energy differences of free ions is 5640 cm−1/electron.

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ASSOCIATED CONTENT

S Supporting Information *

Additional figure showing DOS of Na24Y23Ce1F96 when the f electron is at spin(α) and spin(β). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by South Dakota Governor’s Office of Economic Development, NSF award EPS0903804, and by DOE, BES − Chemical Sciences, NERSC Contract No. DE-AC02-05CH11231, allocation Award 85213, 86185 “Computational Modeling of Photocatalysis and Photoinduced Charge Transfer Dynamics on Surfaces”. The USD High Performance Computing facilities are gratefully acknowledged.

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