Infrared spectroscopic studies of partially deuterated ethanes and the r0, rz, and re structures

Samples of CH₃CH₂D, CH₃CHD₂, CD₃CH₂D, and CD₃CHD₂ have been prepared, and their infrared spectra recorded. Analysis of type B or type C “perpendicular” bands has enabled the rotational parameter ($\text{A}{}_0\text{-}\text{B}{}_0$) to be determined for all four species. These have been combined with existing infrared, Raman, and microwave data for CH₃CH₃, CD₃CD₃, and CH₃CD₃ species, to determine the ground state (r₀) and ground state average (r_z) structures within narrow limits. Zero point energy effects on the average structure are determined to be a CH bond shortening of 0.0015(3) Å and an HCC angle opening of 0.010(5)° on deuteration. These effects enable the equilibrium structure of ethane to be estimated. The r_z(CC) bond length is determined to be 1.5351(2) Å, which is significantly longer than previous estimates involving electron diffraction data.     

Analysis of the (ν1 + ν2) and (ν2 + ν3) combination bands of ozone

The fine structures of the (ν₁ + ν₂) and (ν₂ + ν₃) combination bands of ozone in the 5.7-μm region have been recorded and analyzed. The two vibrational states are coupled through Coriolis and second-order distortion terms. The interaction has been treated by the numerical diagonalization of the secular determinant for the two coupled states. With the centrifugal distortion parameters fixed to the ground state values, the following constants have been obtained: ν₁ + ν₂ = 1796.26₆, A₁₁₀ = 3.610₄, B₁₁₀ = 0.4414₅, $\text{C}{}^1{}_{110}\text{= 0.3902}{}_9$, ν₂ + ν₃ = 1726.52₆, A₀₁₁ = 3.553₇, B₀₁₁ = 0.4398₂, $\text{C}{}^1{}_{011}\text{= 0.3884}{}_4$, Y₁₃ = ?0.46₆, and X₁₃ = ?0.01₀ cm^?1. In addition, the following anharmonic constants have been obtained: x₁₂ = ?7.82₁ and x₂₃ = ?16.49₄ cm^?1. The value of the dipole moment ratio, $\text{R =}\text{〈011|μ}{}_{\mathrm{z}}\text{|0〉}\text{〈110|μ}{}_{\mathrm{x}}\text{|0〉}$, is 1.30 ± 0.10.     

The Ã1B2-X̃1A1 two-photon fluorescence excitation spectrum of 1,3-difluorobenzene

The $\text{A}\text{?}{}^1\text{B}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ system of 1,3-difluorobenzene has been observed using the technique of two-photon fluorescence excitation obtained with a pulsed dye laser. Calibration was achieved by a combination of the neon optogalvanic spectrum and etalon fringes. In circular, compared to linear, polarization the bands divide into two groups, those which are B₂-A₁ and which retain their intensity with circular polarization, and those which are A₁-A₁ and lose about 60% of their intensity under the same conditions. These two kinds of bands also show characteristic rotational contours. All of the A₁-A₁ bands whose assignments are established obtain their intensity through vibronic interaction in which the vibration ν′₂₅ (ν′₁₄ in the Wilson numbering) mixes the $\text{A}\text{?}$ with, presumably, the $\text{X}\text{?}$ state. There is an important Fermi resonance between the 9¹ and 10¹11¹ levels. Parts of the one-photon absorption spectrum have been photographed to identify sequences associated with the 0₀⁰ band for comparison with those observed in the two-photon spectrum, and to search for bands involving odd quanta of b₂ vibrations, including ν′₂₅ (ν′₁₄); none was found.     

On the electronic spectrum of the Mo2 molecule observed after flash photolysis of Mo(CO)6

Absorption and emission spectra of Mo₂ were investigated using flash photolysis of the Mo(CO)₆ molecule. Tentative vibrational and rotational analyses of the ⁹⁸Mo₂ spectra were performed. For the ground state, ${}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ type was proposed with ω_e = 477.1 cm^?1, $\text{r}{}_{\mathrm{e}}\text{= 1.929}\text{A}\text{?}$, and D₀(Mo₂) = 95 ± 15 kcal mole^?1. The results were compared with theoretical calculations for Mo₂ and experimental results for Cr₂ obtained previously. It seems reasonable that the transition metal diatomic molecules of this type have a high bond order.     

Selenium dioxide: Rotational analysis and Franck-Condon calculations for the 3130 Å 1B2-X̃1A1 absorption system

Band contour analyses of the absorption bands of ⁷⁸Se¹⁶O₂ and ⁸⁰Se¹⁶O₂ at 2949 Å, assigned to the 1₀³ transition (King and McLean, in press), show that they are type A, with transition moment directed in-plane and parallel to the line joining the oxygen nuclei. The electronic transition responsible for the B absorption system of the molecule is therefore ${}^1\text{B}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ under the C_2v point group. The contour analysis gives the excited state bond angle as 101.0°, and the bond length as 1.74 Å. The latter value is confirmed by Franck-Condon calculations. There is therefore an increase in bond length and a decrease in bond angle upon electronic excitation. This agrees with the predictions of molecular orbital theory.     

Determination of molecular constants of nitric oxide from (1-0), (2-0), (3-0) bands of the 15N16O and 15N18O isotopic species

The (1-0), (2-0), and (3-0) transitions of ¹⁵N¹⁶O and ¹⁵N¹⁸O are investigated. The wavenumbers of the rotation-vibration lines are reported for the overtone bands and the ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ (1-0) subband. It is shown that in the data reduction it is advantageous to calculate first merged spectroscopic constants ignoring the Λ-type doubling. The vibrational constants ω_e, ω_ex_e, ω_ey_e and the vibrational dependence of the rotational constants are determined. The study of ¹⁵N¹⁸O allows the determination of the equilibrium values of the centrifugal distortion correction A_De to the spin-orbit constant and of the spin-rotation constant γ_e from the isotopic invariance of the ratios $\text{A}{}_{\mathrm{<mtext>De</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}$ and $\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}$. It is found that $\text{A}{}_{\mathrm{<mtext>De</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}\text{= (?3.9 ± 1.3) × 10}{}^{\mathrm{?6}}$ and $\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}\text{= (?4.00 ± 0.05) × 10}{}^{\mathrm{?3}}$.     

Application of Watson's centrifugal distortion theory to water and light asymmetric tops. General methods. Analysis of the ground state and the ν2 state of D2O16

Watson's reduced Hamiltonian proved adapted for treating the rotation of light molecules which show very large effects of centrifugal distortion. In order to determine which terms are required and which are negligible (choice of an effective model), we have studied the order of magnitude of the terms from a theoretical point of view. This order depends on the coefficient and the operator. It is shown (1) that the order of the coefficients depends not only on the degree of the components of the angular momentum but also on the asymmetry of the molecule and (2) that the factor due to the operator depends on the asymmetry and on the rotational quantum numbers J and K_?1. It is established that diagonal and off-diagonal terms are of similar importance for small K_?1 while, for large K_?1, the diagonal terms are the leading ones. Further, the number of terms of high degree may be reduced by numerically telescoping the series. The terms to be retained are best determined by a statistical analysis. Terms resulting from telescoping completely or partially lose their original meaning.The previous theory has been applied to the ground state and to the ν₂ state of D₂O¹⁶. Thirteen new microwave lines are reported in this paper. In the rotational analysis, microwave and infrared data were used simultaneously: 22 constants could be determined for the groundstate and only nine for the ν₂ state. The constants related to the z-axis are most affected in the latter. The A, B, C constants corresponding to the standard Hamiltonian, i.e., the Kivelson-Wilson Hamiltonian, are, in MHz:

$\text{A}{}_{\mathrm{gr}}\text{=462 291.70, B}{}_{\mathrm{gr}}\text{=217 982.29, C}{}_{\mathrm{gr}}\text{=145 301.13, A}{}_{\mathrm{v2}}\text{=498 746, B}{}_{\mathrm{v2}}\text{=219 960, C}{}_{\mathrm{v2}}\text{=143 672.}$

     

Energy levels of low lying triplet states of N2 from infrared emission studies

A theoretical model used to describe the $\text{B′}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}$ and B³Π_g states of N₂ is presented. Using recently acquired high resolution spectra of the $\text{B′}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{→ B}{}^3\text{Π}{}_{\mathrm{g}}$ (0-0) band, rotational energy levels of the v = 0 vibrational levels of these two states are generated with this model. These levels are in excellent agreement with those obtained using a combination differences technique. The precision of the model generated levels is 0.01 cm^?1. The previously unpublished rotational levels of Dieke and Heath for the $\text{A}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$, B³Π_g and C³Π_u states are referenced to the $\text{N}{}_2\text{X}{}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ (v = 0, J = 0) ground level and tabulated here. Estimates of the precision of their work are made.     

Analysis of the perturbed NO22B2 ← 2A1 system in the 591.4- to 592.9-nm region based on sub-Doppler laser spectroscopy

Sub-Doppler excitation spectra of NO₂, covering four vibronic bands within the spectral range from 16 861 to 16 903 cm^?1, were measured with a resolution of down to 10 MHz in a collimated supersonic molecular beam. Unambiguous assignment of all prominent lines in the 42-cm^?1-wide interval of the ${}^2\text{B}{}_2\text{←}{}^2\text{A}{}_1$ excitation spectrum was achieved by recording for each excitation line at least four vibrational bands of the corresponding fluorescence spectrum with completely resolved rotational lines. From least-squares fits to the line positions in the excitation spectra the rotational, fine, and hyperfine structure of the ²B₂ state was analyzed. A perturbation analysis, based on information from both types of spectra, confirms earlier models of vibronic coupling with high-lying vibrational levels of the ²A₁ ground state and gives evidence for spin-orbit coupling. Possible models are discussed which may explain the observed perturbations.     

The high resolution infrared spectrum of the ν1 band of HOCl

The ν₁ bands of HO³⁵Cl and HO³⁷Cl have been recorded. Both the A- and B-type rotational transitions of these hybrid bands have been completely assigned, and spectroscopic constants have been obtained for both the ground and upper state. The ratio of the electric dipole moment derivatives $\text{(}\text{?μ}{}_{\mathrm{a}}\text{?Q}{}_1\text{)}\text{(}\text{?μ}{}_{\mathrm{b}}\text{?Q}{}_1$ has been found to be 0.985 ± 0.05 for ν₁.     

Thioformaldehyde: Vibrational analysis of the Ã1A2-X̃1A1 visible absorption system

The electronic absorption spectra of thioformaldehyde and thioformaldehyde-d₂ have been obtained. A vibrational analysis of the discrete band system in the 6100-4400-Å region is reported. The type A origin bands are at $\text{16 394}\text{16 484}\text{cm}{}^{\mathrm{?1}}$ for $\text{CH}{}_2\text{S}\text{CD}{}_2\text{S}$, and are magnetic dipole allowed. The electronic transition is $\text{A}\text{?}{}^1\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ under the C_2v point group. Most of the intensity of the system is in type B bands, and is due to vibronic mixing with higher ¹B₂ states when the inversion mode ν₄ is excited. The molecule in the excited ¹A₂ state is “floppy-planar,” having a broad potential function with a barrier of the order of 20 cm^?1 to the inversion motion.     

Transition moment variation in the A2Σ+ → X2Πi transition of HCl+, DCl+ and HBr+

Relative emission intensities of sixteen bands of HCl⁺ (A²Σ⁺ - X²Π_i), four bands of DCl⁺ (A²Σ⁺ - X²Π_i), and 5 bands of HBr⁺ (A²Σ - X²Π_i) have been made using both ion-beam excitation and microwave discharge sources. Intensities were determined by comparison with computer-generated spectra. Treatment of the data within the r-centroid approximation shows that in HCl⁺ the electronic transition moment decreases strongly at large $\text{r}{}_{\mathrm{v\prime v″}}$ [$\text{Re}\text{α}\text{exp}\text{(?3.6}\text{r}{}_{\mathrm{v\prime v″}}\text{)}\text{for 1.44}\text{A}\text{?}\text{< r}{}_{\mathrm{v\prime v″}}\text{< 1.82}\text{A}\text{?}$] but levels off at shorter $\text{r}{}_{\mathrm{v\prime v″}}$. DCl⁺ data agree quantitatively with HCl⁺. The variation in the HBr⁺ moment is similar, with $\text{R}\text{e}\text{α}\text{exp}\text{[?4.5}\text{r}{}_{\mathrm{v\prime v″}}\text{]}\text{for 1.58}\text{A}\text{?}\text{<}\text{r}{}_{\mathrm{v\prime v″}}\text{< 1.78}\text{A}\text{?}$.     

Infrared bands of 12C2HD

The rotational structure of about 40 bands of ¹²C₂HD observed in the region 6000?600 cm^?1 has been measured and interpreted with the purpose of determining a comprehensive set of molecular constants for this isotopic variety of acetylene. Combining these data with the results for ¹²C₂H₂ and ¹²C₂D₂, a reevaluation of the equilibrium internuclear distances for the acetylene molecule has been made: $\text{r}{}_{\mathrm{e}}\text{(CH) = 1.06215 ± 17 × 10}{}^{\mathrm{?5}}\text{A}\text{?}$ and $\text{r}{}_{\mathrm{e}}\text{(CC) = 1.20257 ± 9 × 10}{}^{\mathrm{?5}}\text{A}\text{?}$ were obtained. This paper presents all the molecular constants derived in this study.     

Thioformaldehyde: Rotational analyses of the Ã1A2-X̃1A1 visible absorption system

The $\text{A}\text{?}{}^1\text{A}{}_2\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ electronic absorption spectra of CH₂S and CD₂S have been photographed under high resolution. Selected bands have been rotationally analyzed by least squares line fitting and by band contour methods. Improved rotational constants have been obtained for the ground states of CH₂S and CD₂S by use of combination differences. Bands of all three polarizations appear in the electronic spectrum. The type A origin band is magnetic dipole allowed, whereas the 4₀¹ band is type B. Perturbations are identified in the 0₀⁰ and 3₀¹4₀³ bands of CH₂S. The rotational constant A in the upper state decreases rapidly, in accordance with theoretical calculations, as successive quanta of the inversion mode ν′₄ are excited. The planar inertial defect has a small positive value in the zero level of the upper state although the molecule is slightly nonplanar; the r₈ geometry is $\text{r(}\text{CH}\text{) = 1.082}\text{A}\text{?}$, $\text{r(}\text{CS}\text{) = 1.701}\text{A}\text{?}$, angle HCH = 120°, and the out-of-plane angle is approximately 10°.     

Formaldehyde as a semirigid bender: The rotation-inversion spectrum in its Ã1A2 excited state

The semirigid bender Hamiltonian [Bunker and Landsberg, J. Mol. Spectrosc.67, 374–385 (1977)] was used to fit the rotation-inversion energy level separations in the $\text{A}\text{?}{}^1\text{A}{}_2$ excited state of formaldehyde. We fix the r⁰(CH) bond length and allow the R(CO) bond length and (H?H) bond angle to vary with the inversion angle ρ. The fit to 64 rotation-inversion energies (with v₄ and J < 4) is significantly better with a standard deviation of 0.199 cm^?1 than when the rigid bender [Bunker and Stone, J. Mol. Spectrosc.41, 310–332 (1972)] is used. The barrier height to planarity is 358 cm^?1 and the equilibrium ρ_e = 34.7°. The CO bond length is found to decrease by 0.034 from 1.3670 Å and the H?H angle by about 6 from 122.4° as the molecular configuration changes from planar to pyramidal. The rigid bender model developed earlier by Moule and Rao for formaldehyde [J. Mol. Spectrosc.45, 120–141 (1973)] is then used to fit the 32 rotation-(out-of-plane) bending energy levels (with v₄ = 0 and 1) of the $\text{X}\text{?}{}^1\text{A}{}_1$ ground electronic state of H₂CO. For this, a simple potential consisting of quadratic and quartic terms is used and the standard deviation of the fit is 0.148 cm^?1.     

The 370-nm electronic spectrum of tropolone: Evidence from single vibronic level fluorescence spectra regarding the assignment of some vibrational fundamentals in the X̃ and à states

Single vibronic level fluorescence (SVLF) spectra of tropolone from vibronic levels in the $\text{A}\text{?}{}^1\text{B}{}_2$ electronic state, in combination with recently reported supersonic jet spectra, have enabled the assigning of many absorption bands in the region of 0₀⁰ which had previously been impossible. Some of the complexity in these bands has been shown to be due to a large Duschinsky effect involving the two lowest b₁ vibrations, ν₂₅ and ν₂₆. It has been shown that these vibrations have wavenumbers of 176 and 110 cm^?1, respectively, in the $\text{X}\text{?}$ state, and 172 and 39 cm^?1 in the $\text{A}\text{?}$ state. This last value shows how unresistent the molecule is in the $\text{A}\text{?}$ state to out-of-plane bending in the region of the five-membered ring. Other aspects of the vibrational complexity are due to the effect of ν′₂₆ in increasing the barrier to tunnelling of the hydrogen-bonding proton in the $\text{A}\text{?}$ state contrasting with very little effect of ν″₂₆ in the $\text{X}\text{?}$ state.     

The spectrum of HeNe+

Discharges through mixtures of helium and neon show two band groups near 4250 and 4100 Å as first observed by Druyvesteyn. These bands, assigned to the HeNe⁺ ion by Tanaka, Yoshino, and Freeman, have been studied under high resolution and have been fairly completely analyzed. The upper state of the transition is a very weakly bound state resulting from $\text{He}{}^{\mathrm{+}}\text{(}{}^2\text{S) +}\text{Ne}\text{(}{}^1\text{S}{}_0\text{)}$. There are two lower states resulting from the two components of $\text{Ne}{}^{\mathrm{+}}\text{(}{}^2\text{P) +}\text{He}\text{(}{}^1\text{S}{}_0\text{)}$. The upper of these two (${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$) is also very weakly bound while the lower of the two, the ²Σ⁺ ground state, has a dissociation energy of 0.69 eV and an r_e value of 1.30 Å. All bands in both band groups show four branches designated R_ff, Q_ef, Q_fe, and P_ee. From their analysis the rotational constants in the various vibrational levels of the three electronic states have been determined. While no spin splitting in the B²Σ⁺ state has been found the ground state X²Σ shows a very large spin splitting and the $\text{A}{}_2{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state a very large $\text{Ω-type}$ doubling. The vibrational numberings in all these states were established by the study of the spectrum of ³HeNe⁺. At the same time the hyperfine structure observed in all lines of ³HeNe⁺ confirmed the nature of the upper state B²Σ⁺ as resulting from He⁺ + Ne, i.e., by charge exchange from the ground state. The ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ component of the ²Π state has not been observed, presumably because of low intensity.     

Effect of the internal rotation of the CHD2 group on the CH stretching infrared and Raman spectra of toluene C6D5CHD2 and nitromethane NO2CHD2 in vapor phase

Gas-phase infrared and Raman spectra of toluene C₆D₅CHD₂ and nitromethane NO₂CHD₂ were recorded in the CH stretching region. They are all characterized by a strong band with a weaker one at lower frequency. These bands have simple Raman profiles and their infrared contours are respectively of A and C type. A quantum theory of these spectra is put forward, assuming an anharmonic coupling of the νCH mode with the internal rotation of the CHD₂ group in the adiabatic approximation. Theoretical calculations based on this model give a good fit of the experimental Raman bandshapes and a good picture of the observed infrared spectra. Thus each of the observed bands can be characterized. The frequency of the intense band is the average of that of the νCH mode during the almost free internal rotation of the CHD₂ group, while the frequency of the weaker band is approximately equal to the minimal νCH frequency. This last one corresponds to the position of the CH bond in a plane perpendicular to that of the molecule (ν_⊥CH). The frequency difference between ν_∥CH (the CH bond being in the plane of the molecule) and ν_⊥CH is found to be 42 cm^?1 for the two compounds.     

The electronic isotope shift in the A2Π-X2Σ+ bands of BeH,BeD and BeT

The 0-0, 1-1, 2-2, and 3-3 bands of the A²Π-X²Σ⁺ transition of the tritiated beryllium monohydride molecule have been observed at 5000 Å in emission using a beryllium hollow-cathode discharge in a He + T₂ mixture. The rotational analysis of these bands yields the following principal molecular constants.

$\text{A}{}^2\text{Π:}\text{B}{}_{\mathrm{e}}\text{= 4.192}\text{cm}{}^{\mathrm{?1}}\text{; r}{}_{\mathrm{e}}\text{= 1.333}\text{A}\text{?}$

$\text{X}{}^2\text{Σ:}\text{B}{}_{\mathrm{e}}\text{= 4.142}\text{cm}{}^{\mathrm{?1}}\text{; r}{}_{\mathrm{e}}\text{= 1.341}\text{A}\text{?}$

$\text{ω}{}_{\mathrm{e}}\text{′ ? ω}{}_{\mathrm{e}}\text{″ = 16.36}\text{cm}{}^{\mathrm{?1}}\text{; ω}{}_{\mathrm{e}}\text{′X}{}_{\mathrm{e}}\text{′ ? ω}{}_{\mathrm{e}}\text{″X}{}_{\mathrm{e}}\text{″ = 0.84}\text{cm}{}^{\mathrm{?1}}$

From the pure electronic energy difference (E_Π - E_Σ)_BeT = 20 037.91 ± 1.5 cm^?1 and the corresponding previously known values for BeH and BeD, the following electronic isotope shifts are derived

$\text{ΔE}{}_{\mathrm{e}}{}^{\mathrm{i}}\text{(}\text{BeH}\text{?}\text{BeT}\text{) = ?4.7 ≠ 1.5}\text{cm}{}^1\text{, ΔE}{}_{\mathrm{e}}{}^{\mathrm{i}}\text{(}\text{BeH}\text{?}\text{BeT}\text{) = ?1.8 ≠ 1.5}\text{cm}{}^1$

and related to the theoretical approach given by Bunker to the problem of the breakdown of the Born-Oppenheimer approximation.     

Overtone bands of 14N16O and determination of molecular constants

The wavenumbers of the rotation-vibration lines of ¹⁴N¹⁶O are reported for the (2-0) and (3-0) bands. The full set of spectroscopic constants for the three bands (1-0), (2-0), and (3-0) has been determined with the method developed by Albritton, Schmeltekopf, and Zare for merging the results of separate least-squares fits. The vibrational constants ω_e, ω_ex_e, ω_ey_e, and the vibrational dependence of the rotational constants have been deduced. The apparent spin-orbit constant $\text{A}\text{?}{}_{\mathrm{<mtext>v</mtext>}}$ and its centrifugal correction $\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext>}}$ (including the spin-rotation constant) have a vibrational dependence of the following form: $\text{A}\text{?}{}_{\mathrm{v}}\text{=}\text{A}\text{?}{}_{\mathrm{e}}\text{? α}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{) + γ}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{)}{}^2$ and $\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext><mtext>v</mtext>}}\text{=}\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext><mtext>e</mtext>}}\text{? β}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{) + δ}{}_{\mathrm{A}}\text{(v +}\text{built+1}\text{2}\text{)}{}^2$; the values of the constants in these two equations have been determined.