Non-cubic CaAl 2. Si 2. ))-E(Γ(p z

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1 Supplementary Figures Cubic FCC Non-cubic CaAl Si -type structure Γ(p xy,,z) Valence Band Maximum Crystal field effect Γ(p z ) Γ(p xy ) Δ =E(Γ(p x,y ))-E(Γ(p z )), Spin orbit coupling E F Δ Δ SO (SO) Supplementary Figure 1. Schematic diagram of the evolution from p x,y,z orbitals into valence bands at Γ point in CaAl Si -type Zintl compounds. The energy difference between top electronic state split from p x,y band and the electronic state from p z band is defined as Δ(SO). The spin orbit splitting energy of p x,y band is labeled as Δ SO. 1

2 a Energy (ev) Mg Sb DOS (states ev -1 cell -1 ).0 b s p y p x p z d xy d yz d z d xz d x -y Energy (ev) SrZn Sr Zn Sb Γ M K Γ A L H A DOS (states ev -1 cell -1 ) Supplementary Figure. (a) Orbital-projected density of states of Mg 3. (b) Calculated orbital-projected band structures and density of states of representative CaAl Si -type Zintl compound SrZn with positive crystal field splitting energy Δ. p x, p y, and p z orbitals of Sb anions are projected on the band structure. Curve width indicates the relative weight of the component.

3 a α σ/τ (10 14 μw cm -1 K - s -1 ) 15 Mg 3 p = 10 0 cm K CaZn Δ (ev) b α σ (μw cm -1 K - ) K Δ (ev) Supplementary Figure 3. (a) Calculated α σ/τ (p = cm -3 ) and b) experimental power factors 1-16 (α σ) at 300K as a function of the crystal field splitting energy Δ in CaAl Si -type Zintl compounds. The solid lines in Supplementary Fig. 3a represent fitted curves using a B spline. The curve in Supplementary Fig. 3b is guide to the eye, showing the maximum values corresponding to optimal carrier concentrations. Solid solutions YbCd 1.6 Zn 0.4 and EuZn 1.8 Cd 0. with zt values,3 above unity are marked in red. 3

4 a α σ/τ (10 14 μw cm -1 K - s -1 ) p = 10 0 cm K b α σ/τ (10 14 μw cm -1 K - s -1 ) Optimal p 300 K Δ(SO) (ev) Δ(SO) (ev) Supplementary Figure 4. Calculated room temperature theoretical α σ/τ including spin orbit coupling effect at a given carrier concentration p = cm -3 (a) and optimal p (b) (10 18 ~10 1 cm -3 ) as a function of Δ(SO) in representative CaAl Si -type Zintl compound Mg 3. The red lines represent fitted curves using a B spline. 4

5 a α σ/τ (10 14 μw cm -1 K - s -1 ) optimal p p = cm -3 p = 10 0 cm -3 p = cm -3 p = cm -3 Mg K Δ (ev) b α σ/τ (10 14 μw cm -1 K - s -1 ) Mg K K K Optimal p Δ (ev) Supplementary Figure 5. (a) Calculated α σ/τ of Mg 3 at 300 K versus Δ at varying carrier concentrations (10 19 ~ cm -3 ). The red curve shows the maximum values corresponding to optimal carrier concentrations. Data points for other unoptimized carrier concentrations fill up the whole pink area right below the best curve with optimal carrier concentration. (b) Calculated optimal α σ/τ at 300 K and 600 K as a function of Δ in representative CaAl Si -type Zintl compounds Mg 3 and CaZn. The solid lines represent fitted curves using a B spline. 5

6 a K b K zt 0.1 zt Δ (ev) Δ (ev) Supplementary Figure 6. Experimental zt values 1-18 of CaAl Si -type Zintl compounds at 300 K (a) and 500 K (b) as a function of Δ. Mixtures or solid solutions YbCd 1.6 Zn 0.4 and EuZn 1.8 Cd 0. with zt values,3 above unity at high temperature are marked in red. The curve is guide to the eye, showing the maximum values corresponding to optimal carrier concentrations. 6

7 a μ 0 (m*/me) 3/ (cm V -1 s -1 ) K Δ (ev) b zt opt K Δ (ev) Supplementary Figure 7. Weighted mobility μ 0 (m * /m e ) 3/ (a) and optimum zt (b) at room temperature as a function of Δ in CaAl Si -type Zintl compounds. The μ 0 (m * /m e ) 3/ values and optimum zt values are simulated based on experimental data from references,4-7,9-15,18 using the single parabolic band model under acoustic phonon scattering mechanism. Compound YbCd 1.6 Zn 0.4 with zt value above 1 at high temperature is marked in red. The curve is guide to the eye, showing the maximum values corresponding to optimal carrier concentrations. The insufficiency of the single parabolic band model in describing the anisotropic valence bands in CaAl Si type Zintl compounds and the varying carrier concentrations from different references means that the data points for unoptimized carrier concentrations will fill up the whole area below the red curve. 7

8 a Δ (ev) SrZn P SrZn As SrZn CaZn P CaZn CaZn As Sb EuCd CaCd As SrCd EuZn As As CaCd SrCd BaMg Bi BaMg EuZn P YbZn As YbZn YbCd SrMg SrMg Bi EuMg Bi EuZn EuMg YbMg Bi CaMg CaMg Bi YbMg b Δ (ev) SrZn SrZn P EuCd As As SrZn BaMg Bi SrCd CaZn As CaZn P CaZn SrCd P CaCd SrCd As CaCd As EuCd EuCd P YbZn Sb YbCd YbZn EuZn P As SrMg Bi SrMg As EuZn Sb BaMg Sb SrMg EuMg Bi CaCd P EuMg Sb YbMg Bi CaMg CaMg Bi CaMg As YbMg Sb Mg 3 As Mg a (Å) Mg 3 Mg 3 As E SO g (ev) Supplementary Figure 8. Crystal field splitting energy Δ as a function of lattice constant a (a) and calculated energy gap including SOC E SO g (b) in CaAl Si -type Zintl compounds. The dotted line is the zero line. Filled circles are positive Δ compounds. Empty circles are negative Δ compounds. From the solid solution map, we find that there are still lots of potential TE candidates including (Eu,Sr)Zn, (Eu,Ca)Zn, (Eu,Yb)Zn, (Yb,Ba)Mg, Ca(Mg,Cd), Yb(Mg,Zn), (Ca,Ba)Mg Bi, etc. 8

9 Energy (ev) EuZn 1.75 Cd Z Γ F Wave vectors Q Z Supplementary Figure 9. Band structure of solid solution EuZn 1.75 Cd 0.5 with nearly zero Δ value. 9

10 5 EuZn 1.8 Cd 0. EuZn YbCd 1.6 Zn 0.4 YbCd κ (W m -1 K -1 ) 4 3 EuCd YbZn T (K) Supplementary Figure 10. Temperature dependence of thermal conductivity κ of YbZn (EuZn ), YbCd (EuCd ), and their solid solutions YbCd 1.6 Zn 0.4 and EuZn 1.8 Cd 0.,3. 10

11 a α σ/τ (10 14 μw cm -1 K - s -1 ) Mg 3 CaZn p = 10 0 cm K b α σ/τ (10 14 μw cm -1 K - s -1 ) K Mg 3 CaZn ε (%) ε (%) Supplementary Figure 11. Theoretical α σ/τ at a given hole concentration p= cm -3 (a) and at optimal hole concentration (b) at room temperature as a function of biaxial strain ε in two representative CaAl Si -type Zintl compounds, Mg 3 and CaZn. The solid lines represent fitted curves using a B spline. 11

12 Supplementary Figure 1. Calculated orbital-projected density of states of (a) chalcopyrite compound AgGaTe, (b) lithium intercalated metal dichalcogenide LiZrSe, and (c) metal dichalcogenide ZrS. 1

13 Δ (ev) LiZrSe ZrS ε (%) Supplementary Figure 13. Crystal field splitting energy Δ as a function of biaxial strain ε in representative metal dichalcogenide ZrS and lithium intercalated metal dichalcogenide LiZrSe. The solid lines represent fitted curves using a B spline. 13

14 0.6 TaS 0.4 TiSe HfTe Δ (ev) TiS HfSe ZrTe -0. HfS ZrSe -0.4 ZrS SnS SnSe a (Å) Supplementary Figure 14. Crystal field splitting energy Δ as a function of lattice constant a in carefully selected metal dichalcogenides. The red stars with trend line represent mixtures or solid solutions between two metal dichalcogenides with positive and negative Δ values. The dotted line is the zero line. Filled circles are positive Δ compounds. Empty circles are negative Δ compounds. 14

15 a b Energy (ev) α DOS (states ev -1 cell -1 ) Supplementary Figure 15. Electronic properties of EuZn. (a) Density of states around the Fermi level. The black lines are for spin-polarized all-electron calculations and the blue lines (which are identical for the two spin channels) are for non-spin-polarized PAW 19 calculations. (b) Calculated Seebeck coefficient at p = 10 0 cm -3 along the a and c directions. The full lines correspond to all-electron calculations. The Seebeck coefficients of spin-channels have been averaged as α = (α σ + α σ )/(σ + σ ). The dashed lines correspond to non-spin-polarized PAW calculations. 15

16 Supplementary Tables Supplementary Table 1. Room temperature thermoelectric transport properties of CaAl Si -type Zintl compounds,4-7,9-16,18. Density of states effective mass m*, intrinsic mobility μ 0, and weighted mobility μ 0 (m*/m e ) 3/ listed in this table are calculated from Hall data using single parabolic band model under acoustic phonon scattering mechanism. Hall data of Mg 3 Bi marked in red was not used for modeling weighted mobility and optimal zt since Mg 3 Bi is a semimetal indicated by band gap calculation with spin orbit coupling (see Supplementary Table ). Systems YbCd α (μv K -1 ) σ (10 4 S m -1 ) p (10 0 cm -3 ) μ H (cm V -1 s -1 ) m*/m e μ 0 (m*/m e ) 3/ (cm V -1 s -1 ) YbCd 1.6 Zn YbCdZn YbCd 0.8 Zn YbCd 0.4 Zn YbCd Mg YbCd EuCd YbZn (33K) CaZn EuZn CaZn SrZn YbCd EuCd SrZn CaMg Bi YbMg Bi YbMg Bi CaMg Bi YbMg Bi Mg 3 Bi (34K)

17 Supplementary Table. Lattice parameters, calculated crystal field splitting energy Δ, absolute values of theoretical valence band splitting energy with spin orbit coupling (SOC) Δ(SO), and calculated energy gaps E g without SOC and E SO g with SOC of CaAl Si -type Zintl compounds. Potential TE compounds with nontoxic, low cost, and earth-abundant elements are marked in red color. a and c are experimental data from ICSD database 0, while the internal co-ordinates are determined by energy minimization. Lattice parameters and atomic positions of EuZn, EuCd, YbZn, and YbCd are relaxed using HSE06 functional 1 with the aim to keep consistent with solid solution calculations. CaAl Si -type Zintl compounds (Space group P3 _ m1) Compounds a (Å) c (Å) Δ (ev) Δ(SO) (ev) E g (ev) E SO g (ev) CaZn P CaCd P SrZn P SrCd P EuZn P EuCd P Mg 3 As CaMg As CaZn As CaCd As SrMg As SrZn As SrCd As YbZn As EuZn As EuCd As Mg CaMg CaZn CaCd SrMg SrZn SrCd BaMg EuMg EuZn EuCd YbMg YbZn YbCd

18 Mg 3 Bi CaMg Bi SrMg Bi BaMg Bi EuMg Bi YbMg Bi Supplementary Table 3. Calculated lattice parameters, crystal field splitting energy Δ, and calculated energy gap E g without SOC of several MX metal dichalcogenides with P3 _ m1 as space group. ZrS and TiS marked in red are two constituent compounds used for the validation of solid solution map (Supplementary Fig. 14). Lattice parameters and atomic positions of ZrS and TiS are relaxed using the same method as solid solution calculations. MX metal dichalcogenides (Space group P3 _ m1) Compounds a (Å) c(å) Δ (ev) E g (ev) TaS ZrS TiS ZrSe SnS TiSe ZrTe SnSe HfS HfSe HfTe Supplementary Table 4. Effective mass along different lattice directions, conductivity effective mass and the calculated carrier scattering time τ for p-type Mg 3. Effective mass is calculated from ab initio band structure. m * ( m ) xx e p-type Mg 3 m * ( m ) m * ( m ) zz e I e τ 14 ( 10 s)

19 Supplementary Notes Supplementary Note 1. Orbital characteristics of valence bands affected by crystal field effect Schematic diagram of electronic bands at valence band maximum in CaAl Si -type Zintl compounds is exemplified in Supplementary Fig. 1. Doubly degenerate band Γ(p x,y ) and nondegenerate band Γ(p z ) are mainly from p x,y and p z orbitals of anions (see Supplementary Fig. ). The seperation of Γ(p x,y ) and Γ(p z ) is caused by the crystal field splitting of p orbitals into p x,y and p z orbitals. The crystal field splitting parameter Δ, is defined as the energy difference between Γ(p x,y ) and Γ(p z ). Experimental reported layered CaAl Si -type Zintl compounds, including YbCd, YbZn, EuZn, EuCd, CaZn, SrZn, CaMg Bi, YbMg Bi, Mg 3 Bi, Mg 3, Eu(Zn 1-x Cd x ), Yb(Zn 1-x Cd x ), and Yb 0.75 Eu 0.5 Cd (calculated Δ = 0.04 ev) are studied and discussed in this work. For Yb and Eu elements, the PAW 19 potentials, in which f-electrons are kept frozen in the core, are used for first principles calculations. To validate this approach we performed spin-polarized calculations on EuZn using WIENk. The mbj potential 3 was applied and a U = 10 ev was added for the Eu 4f-states as well as a U = 4 ev for the Zn 3d-states. Supplementary Figure 15 illustrates how the localized nature of the Eu 4f-states leads to a very small magnetic splitting around the Fermi level. Consequently, the calculated transport coefficients with the two methods (PAW f -frozen core and all-electron spin polarized) are in very good internal agreement. Supplementary Note. The effect of spin orbit coupling As indicated in Supplementary Fig. 1, spin orbital coupling (SOC) will further split doubly degenerate p x,y energy level into two electronic states: one still lies at the valence band maximum; the other will shift downwards and might be far away from band edge depending on the degree of spin orbit coupling. Although SOC further lifts the degeneracy of p x,y band which will degrade the electrical transport properties, there still exists two electronic states at valence band maximum, each of which possesses the degeneracy of (include spin), that can also be tuned by solid solution method and strain engineering. Since valence band maximum of CaAl Si -type Zintl compounds is mainly from anions, valence band splitting is mainly determined by spin orbit coupling of anion elements. The energy difference between top electronic state split from p x,y band and the electronic state from p z band is defined as Δ(SO). The spin orbit splitting energy of p x,y band is labeled as Δ SO. The calculated absolute values of Δ(SO) ( Δ(SO) ) of CaAl Si -type Zintl compounds are listed in Supplementary Table. We find that Δ(SO) and Δ values of most compounds containing relatively heavy elements Sb and Bi (with large Δ SO > 0.6 ev) discussed in this paper are comparable. It means that strong SOC will push one state far away from band edge 19

20 but will only have minor effects on the valence band splitting energy Δ(SO) at band edge. Moreover, for compounds with light elements SOC only has minor effects on the electronic bands at band edges. For this situation, it is reasonable to only consider crystal field effect for electronic structure calculations. In order to check the effect of SOC on electronic transport properties, the electronic structure and transport calculation including SOC effect were carried out for Mg 3 by combining WIENk and BoltzTrap 4. The zone samplings were done with uniform k mesh grids, while a cut-off for the basis set RK max = 9 was used. TB-mBJ functional 3 was adopted for electronic structure calculations. The dependence of calculated power factor including SOC at room temperature on valence band splitting energy Δ(SO) is shown in Supplementary Fig. 4. As expected, theoretical power factor with SOC decreases compared with that without SOC (see Supplementary Fig. 5a) due to lifting the degeneracy of p x,y band; however, it still can be tuned by valence splitting energy Δ(SO) and shows a peak while Δ(SO) approaches zero, fully consistent with the trend only including crystal field effect. The above result proves that SOC effect will not change the main idea of orbital engineering. Therefore, to be simplified it is rational to only include crystal field effect in this work. Supplementary Note 3. Theoretical power factors and their dependence on crystal field splitting The crystal field splitting energy Δ in Fig. a in main text was varied by slightly tuning lattice parameters a and c. Considering the relatively minor changes to the crystal structure for each step, the electron relaxation time τ was assumed to be independent of Δ in this work. For each Δ value, the theoretical power factor α σ/τ at a given hole concentration p = 10 0 cm -3 and optimal hole concentration at different temperatures were calculated and plotted in Supplementary Figs 3a and 5. Supplementary Note 4. Calculation details of solid solutions Solid solution calculations of CaAl Si -type Zintl compounds are carried out in a supercell with 0 atoms ( 1 unit cell). Here we use HSE06 functional 1 for crystal structure optimization. The reason why we use HSE06 functional is that the electronic states at valence band maximum are sensitive to the structural parameters and HSE06 functional can give relatively reasonable structural parameters close to experimental values. Considering the computational cost, a Hellmann-Feynman force convergence criterion of 0.05 ev Å -1 was used. A Monkhorst-Pack k mesh and a Monkhorst-Pack k mesh were used for crystal structure optimization and electronic structure calculations, respectively. Take Eu(Zn 1-x Cd x ) as an example. There are 8 equivalent positions for Zn/Cd atoms, thus the possible atomic concentrations of Cd are multiples of 1/8. The crystal structure of the supercell with the lowest energy is used for further calculations and analysis. The calculation method of electronic structure is the same as mentioned in the main text. Solid solution calculations of layered metal dichalcogenides are carried out in a supercell with 4 atoms ( unit cell). A Monkhorst-Pack k mesh was used for crystal structure 0

21 optimization. Crystal structures are relaxed using PBE functional 5. In layered metal dichalcogenides, interlayer van der waals interaction is very important. Thus, Van der Waals interaction was taken into account during structural relaxation at the vdw-df level with optb86b 6 for exchange functional (optb86b-vdw). Lattice parameters and atomic positions were optimized until the maximum Hellmann-Feynman force acting on each atom is less than ev Å -1. The crystal structure of the supercell with the lowest energy is used for further calculations and analysis. The calculation method of electronic structure is the same as mentioned in the main text. Supplementary Note 5. Calculation details of chalcopyrite AgGaTe, metal dichalcogenides, and lithium intercalated metal dichalcogenide LiZrSe The crystal structures of AgGaTe and LiZrSe were relaxed using HSE06 functional 5. An energy convergence criterion of 10-4 ev and a Hellmann-Feynman force convergence criterion of ev Å -1 were adopted for the calculations of AgGaTe and LiZrSe. For AgGaTe, A Monkhorst-Pack k mesh and a Monkhorst-Pack k mesh were used for crystal structure optimization and electronic structure calculations, respectively. Electronic structure calculation of AgGaTe was implemented by combining the mbj potential and the well-established PBE+U method 3,7, and U = ev was applied on Ag 4d states. The electronic structure calculation method of LiZrSe is the same as that of CaAl Si -type zintl compounds stated in the main text. The crystal structures of layered metal dichalcogenides were relaxed using PBE functional 5. Van der Waals interaction was taken into account during structural relaxation at the vdw-df level with optb86b 6 for exchange functional (optb86b-vdw). The electronic structure calculation method of metal dichalcogenides is the same as that of CaAl Si -type zintl compounds. Supplementary Note 6. Calculation details of carrier scattering time for zt under biaxial strain effect Due to the anisotropic band at valence band maximum, the carrier s scattering time τ is estimated from the relation μ = eτ, where μ is the carrier mobility, * / mi * m I is inertial or conductivity effective mass, and e is the elementary charge. In trigonal p-type Mg 3, * m I can be expressed as m = 3/(/ m + 1/ m ), where * * * I xx zz * m xx and * m zz are effective mass along x and z directions, respectively. Effective mass tensor * mij is defined as m = [ E( k)/ k k ]. * 1 ij i j band structure and thereby * m xx and m * zz of p-type Mg 3 are evaluated from ab initio * m I is calculated using the above formula (see Supplementary Table 4). Using experimental reported mobility 1 μ = 30 cm V -1 s -1 and calculated 1 * m I, we can

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