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J. Phys. Chem. B 2008, 112, 16214–16219
Investigation of 2P Be- Shape Resonances Using a Quadratically Convergent Complex Multiconfigurational Self-Consistent Field Method† Kousik Samanta and Danny L. Yeager* Department of Chemistry, MS-3255, Texas A & M UniVersity, College Station, Texas 77843-3255 ReceiVed: August 5, 2008
We develop, implement, and apply a quadratically convergent complex multiconfigurational self-consistent field method (CMCSCF) that uses the complex scaling theorem of Aguilar, Balslev, and Combes within the framework of the multiconfigurational self-consistent field method (MCSCF) in order to theoretically investigate the resonances originated due to scattering of a low-energy electron off of a neutral or an ionic target (atomic or molecular). The need to scale the electronic coordinates of the Hamiltonian as prescribed in the complex scaling theorem requires the use of a modified second quantization algebra suitable for biorthonormal spin orbital bases. In order to control the convergence to a stationary point in the complex energy hypersurface, a modified step-length control algorithm is incorporated. The position and width of 2P Be- shape resonances are calculated by inspecting the continuum states of Be-. To our knowledge, this is the first time that CMCSCF has been directly used to determine electron-atom/molecule scattering resonances. We demonstrate that both relaxation and nondynamical correlation are important for accurately describing shape resonances. For all of the calculations, the quadratically convergent CMCSCF was found to converge to the correct stationary point with a tolerance of 1.0 × 10-10 au for the energy gradient within 10 iterations or less. 1. Introduction The multiconfigurational self-consistent field method (MCSCF) has been shown to be a very efficient bound-state electronic structure calculation method in accounting for nondynamical and some dynamical correlations very accurately.1-7 With MCSCF several important electronic configurations are present and both the orbitals and state expansion coefficients are optimized simultaneously.1,2 For the resonances originated during the scattering of a low-energy electron off of a neutral or an ionic target (atomic or molecular) that has nondynamical correlation, it is reasonable to use a MCSCF-based method rather than one based on a single configuration Hartree-Fock state. The fact that the resonances lie in the continuum part of the Hamiltonian along with numerous other nonresonant scattering states creates a problem in the direct application of a typical bound-state method. However, the complex scaling theorem developed by Aguilar, Balslev, and Combes empowers a boundstate method to overcome this challenge.8-10 In this study, we develop, implement, and apply complex multiconfigurational self-consistent field method (CMCSCF) utilizing the complex scaling theorem under the framework of the multiconfigurational self-consistent field method (MCSCF). According to the complex scaling theorem, the scattering states appear as the complex eigenvalues of the Hamiltonian when its electronic coordinates are scaled by a complex parameter
η ) R exp(iθ) (R, θ ∈R)
(1)
causing the continua to rotate by an angle of -2θ at each threshold.8-11 However, the eigenvalues corresponding to the † Part of the “Karl Freed Festschrift”. * To whom correspondence should be addressed. E-mail: yeager@ mail.chem.tamu.edu. Telephone: +1-979-845-3436. Fax: +1-979- 845-4719.
bound states and ionization and excitation thresholds remain real and unmodified. Once uncovered, the eigenvalue corresponding to a resonance, E(η), is invariant to changes in η. This stability, described mathematically as
[ ] dnE(η) dηn
) 0; η ) 1, 2, 3, ...
(2)
θ>θc
where the rotation η ) ηc ) Rc exp(iθc) corresponds to the uncovering of the resonance, which makes the resonance stand out among a myriad of nonresonant scattering states which do not show such behavior.8-11 In this region, the eigenfunction corresponding to the resonance is square-integrable, making it feasible for the bound-state methods to probe the resonance.11-27 However, in practical calculations, where we must restrict ourselves to the choice of a finite basis set, only a quasi-stability, given by
dE(η) )0 dη
(3)
is attainable in a very small neighborhood of ηc.11 The resonance position (Er ∈ R) and width (Γ ∈ R) may be obtained from the resonance energy
E(ηc) ) Er - iΓ/2
(4)
There are at least two other successful approaches, quite different from the complex scaling technique outlined above, to investigate resonances by means of using the L2 space of the Hamiltonian in the continuum, namely, the stabilization technique28,29 and the complex absorbing potential (CAP) technique.30-33 In this study, we focus solely on implementing
10.1021/jp806998n CCC: $40.75 2008 American Chemical Society Published on Web 11/07/2008
Investigation of 2P Be- Shape Resonances
J. Phys. Chem. B, Vol. 112, No. 50, 2008 16215
a MCSCF-based method that utilizes the complex scaling technique. Those techniques as well as the complex scaling based methods require basis sets large enough to adequately describe the continuum region of interest. Once a basis set is chosen, resonance positions and widths can be determined with the complex scaling method, However, the stabilization technique requires, in addition, positioning of the barrier and cannot determine widths. With the CAP procedure, the absorbing potential must also be devised and positioned. In the next section, we outline the relevant theory for CMCSCF. Then, we present our computational results for 2P Be - shape resonance in section 3, and in the last section, we conclude and summarize.
and
Sˆ )
∑ Sn0Rˆn
with Rˆn ) |n〉〈0* |-|0〉〈n* |
In eq 10, the states {|n〉} belong to the orthogonal complement space of |0〉.34 In practice, the coefficients {κpq(p > q)} and {Sn0(n*0)} are packed as column vectors (κ_ and S_, respectively), which are again arranged in another column vector known as a step-length vector,
λ)
2. Theory The complex scaling transformation makes the Hamiltonian complex symmetric and non-Hermitian,8-10 which causes its eigenfunctions, {ψi}, to be biorthonormal
〈ψi* |ψj〉
) δij
〈φ*p|φq〉 ) δpq
ˆ )H
∑ [η-2 21 〈φ*p|∇2|φq〉 + η-1 〈φ*p|Zr-1|φq〉]aTpaq +
1 ⇒ E(λ_) ≈ q(λ_) ) E(0 _) + _λTF _ + _λTG _ λ_ 2 _
(13)
ˆ |0〉 E(0 _) ) 〈0* |H
(14)
where
F _ is the first derivative vector given by
F _)
∑
1 〈φ*(1)φ*q(2)|| b r1 - b r 2 | -1 |φr(1)φs(2)〉aTp aTq asar (7) 2 p,q,r,s p
in atomic units, where {aTp } and {ap} are the creation and annihilation operators, respectively, defined in the modified34 second quantization algebra35 for biorthonormal spin orbital bases. The operator aTp creates an electron in the spin orbital φp in the ket if it is not occupied with necessary sign changes to account for the Pauli Principle and gives rise to zero otherwise, whereas ap removes an electron from φp in the ket if it is occupied with again necessary sign changes to account for the Pauli Principle and gives rise to zero if it is not. The roles of aTp and ap are switched when they act on a bra.34 Yeager et al.34 developed a quadratically convergent complex multiconfigurational self-consistent field method (CMCSCF), that is, MCSCF, using the complex scaled Hamiltonian, which may be described in terms of an orthogonal transformation of the state |0〉 to the optimized state |0˜〉34
exp(Sˆ) exp(κˆ )|0〉 ) |0˜〉
(11)
( ) F _κ F _S
(15)
with the elements
p,q
η-1
_κ S_
1 ˆ |0˜〉 ) E(0_) + _λTF E(λ) ) 〈0˜* |H _ + _λTG _ λ_ + . . . (12) 2 _
(6)
The complex scaled electronic Hamiltonian under the BornOppenheimer approximation may be written for an atomic system with nuclear positive charge Z as34
()
The total energy of the optimized state may be written as a Taylor series
(5)
where the asterisk means complex conjugation and the eigenfunctions are normalized to unity.11,34 This biorthonormality is inherited by the spin orbitals, {φp}, which define the Slater determinants that are used to make up {ψi}:
(8)
ˆ,Q ˆ pq]|0〉 (F _ κ)pq ) 〈0* |[H and
ˆ , Rˆn]|0〉 (F _ S)n ) 〈0* |[H
∑ κpqQˆpq p>q
ˆ pq ) aTp aq - aTq ap with Q
(9)
(16)
and G _ is the Hessian matrix given by
G __ κκ G __ κS G __ ) G __ Sκ G __ SS
(
)
(17)
with the following elements
ˆ pq, H ˆ,Q ˆ tu]|0〉 (G __ κκ)pq,tu ) (G __ κκ)tu,pq ) -〈0* |[Q * ˆ ˆ,Q ˆ pq]|0〉 __ Sκ)n,pq ) -〈0 |[Rn, H (G __ κS)pq,n ) (G and * ˆ , Rˆn]|0〉 (G __ SS)m,n ) (G __ SS)n,m ) -〈0 |[Rˆm, H
where
κˆ )
(10)
n*0
(18)
and p > q, r > s, m * 0, and n * 0. Here, [Aˆ, Bˆ, Cˆ] ) (1/2)[Aˆ, [Bˆ, Cˆ]] + (1/2)[[Aˆ, Bˆ], Cˆ] is the symmetric double commutator,
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Samanta and Yeager
where [Aˆ, Bˆ] ) AˆBˆ - BˆAˆ is the commutator and Aˆ, Bˆ, and Cˆ are arbitrary linear operators.34,36 The search for the stationary point is performed by setting the first derivatives of q(λ_) with respect to the elements of the step-length vector ({κpq} and {Sn0}) to zero and then solving for {κpq} and {Sn0}, which leads to the following multidimensional Newton-Raphson equation34,36
λ_ ) -G __ -1F _
(19)
When a calculation is far from the stationary point, the walk on the energy hypersurface must be controlled37 in order to get to the local region and converge to the correct stationary point indicated by the correct number of negative eigenvalues, {gj}, of the Hessian for the θ ) 0 case37 (for θ * 0, Re(gj) values are considered instead),36 where
(Ω __ TG __ Ω __ )ij ) δijgj with Ω __ T ) Ω __ -1
(20)
The step-length control scheme employed here is based on the Fletcher algorithm38 adapted for MCSCF by Jørgensen et al.37 In this study, the scheme is implemented by optimizing q(λ_) subject to the constraint that |λ_| e h, where h is a predetermined small positive number (“trust radius”), or, in other words, optimizing q(λ_) + (ν/2)(λ_Tλ_ - h2), where ν/2 is the Lagrange multiplier (ν ∈ R). This leads to calculating λ_ from the transformed step-length vector, _l ) Ω _ Tλ_, whose elements are
lj(ν) ) -[Re(gj + ν)]-1Re(fj)
(21)
where fj ) (Ω _ TF _ )j, and ν is such that |l_(ν)|| ) h and the number of “shifted” negative eigenvalues of the Hessian, {gj + ν}, is consistent with the state being optimized.36 In a typical electron-atom scattering experiment with a beryllium atom as the target, the creation of a 2P Be- shape resonance may be visualized as the temporary attachment of an incoming electron in one of the p orbitals of the Be atom. With that picture in mind, the search for this resonance may be done by inspecting various excited states (2P) of Be-. In this study, the total energy of the 2P Be- state, E(η), is calculated using CMCSCF as outlined above for a series of η values, and a resonance is found when the total energy does not change with a change in η as guaranteed by eq 3. For η ) ηopt ) Ropt exp(iθopt), which refers to the uncovering of the resonance, eq 3 leads to11
( ∂E∂θ )
Ropt
( ∂E∂η )
) iη
Ropt
)0
(22)
)0
(23)
and
( ∂R∂E )
θopt
)
η ∂E R ∂η
( )
θopt
These equations enable one to find the resonance by inspecting the stability in the θ and R trajectories (i.e., the plots of Im(E) as a function of Re(E) at constant values of R and θ, respectively).39
TABLE 1: Typical Convergence of a Quadratically Convergent CMCSCF Calculation with 2s2p3s3p CAS for Resonance 1 (See Text)a trust largest iteration radius gradient (au) 1 2 3 4 5
0.5 0.5 0.5 0.5 0.5
10-2 + 10-1i 10-4 + 10-3i 10-5 + 10-4i 10-7 + 10-7i 10-11 +10-11i
|λ_|
total energy (au)
10-1 10-2 10-3 10-6 convergedb
-14.6068719834 - 0.0090104430i -14.6065324828 - 0.0089988559i -14.6065336252 - 0.0089991482i -14.6065336254 - 0.0089991477i -14.6065336254 - 0.0089991477i
a Here R ) 0.980, θ ) 0.410 rad, and the basis set is 14s11p (see ref 25). The orbitals at the stationary point of a CMCSCF calculation with the same R and θ ) 0.405 rad are taken as the initial guess for this calculation. b The real and imaginary parts of the largest energy gradient are less than 1.0 × 10-10 au.
3. Results and Discussion While investigating Be- resonances using the complex scaled electron propagator method (single configuration based), Venkatnathan et al.25 found the 14s11p basis set to be the best for this case. Hence, we chose to use the same basis set for our calculations. The Be atom has large nondynamical correlation from considerable mixing of the 1s22p2 configuration with the principal 1s22s2 configuration.4-7 Therefore, for Be-, the choice of the 2s2p complete active space (CAS) is a reasonable starting point. However, in order to correlate 2s and 2p properly, the 2s2p3s3p CAS is a better choice for Be-. In our previous paper,36 we calculated resonance energies by diagonalizing the first block of the M matrix defined in the multiconfigurational spin-tensor electron propagator method (MCSTEP)5 obtained from a CSCF/CMCSCF initial state of the Be atom (this method will be referred to as the M1 method), where we observed that the inclusion of a d function in the CAS has little effect on the position and width of the 2P Be- shape resonance. Hence, we do not report the results with a d orbital present in the CAS here. The incoming electron is very loosely bound to the target. Our previous calculations using the M1 method has shown that the resonant p electron lies mostly outside of the region of the active orbitals.36 In the current calculation, a single electron is placed in a p orbital outside of the CAS, two valence electrons are placed in the CAS, and the 1s orbital is occupied by two core electrons, such that the overall symmetry is 2P. In our previous study using M1 method on the Be atom,36 we have shown that there are two very close-lying low-energy 2P Be- shape resonances. In the current study also, two resonances are obtained with CMCSCF with one and two negative eigenvalues of the Hessian ({Re(gi)}). We shall refer to them as resonance 1 and resonance 2, respectively. No resonance was obtained with zero or more than two negative eigenvalues of the Hessian. Since the CMCSCF method is new, a few words about its convergence are necessary. We observed that the convergence was achieved with a tolerance of 1.0 × 10-10 au for all of the elements of the first derivative vector within 10 iterations or less. Table 1 shows the convergence in a typical quadratically convergent CMCSCF calculation. All of our calculations were done on a cluster of three 64 core Altix 450 machines with Itanium2 Montecito dual core CPUs. The computer time necessary to complete a CMCSCF calculation is very small (e.g., it took 17.70 s of CPU time to complete the calculation in Table 1). It is a common practice to report the resonance energy relative to the total energy of the scattering target. In this paper, we
Investigation of 2P Be- Shape Resonances
J. Phys. Chem. B, Vol. 112, No. 50, 2008 16217 TABLE 2: 2P Be- Shape Resonance Positions and Widths Obtained from Our Calculations resonance 1 width (eV)
position (eV)
width (eV)
direct CSCF/CMCSCF methoda CSCF CMCSCF (CAS: 2s2p) CMCSCF (CAS: 2s2p3s3p)
0.31 0.30 0.31
0.40 0.48 0.49
0.68 0.71 0.73
0.58 1.56 1.58
M1 method initial statesb CSCF CMCSCF (CAS: 2s2p) CMCSCF (CAS: 2s2p3s3p)
0.57 0.57 0.57
1.01 1.15 1.15
0.74 0.73 0.72
1.26 1.14 1.10
method
Figure 1. The θ-trajectories for resonance 1 (see text) using CSCF, CMCSCF with a 2s2p CAS, and CMCSCF with a 2s2p3s3p CAS. The values of R (Ropt) are 1.005, 0.980, and 0.980, respectively. In each of the trajectories, θ started with a value of 0 at the top where Im(ε) ) 0 and incremented linearly in steps of 0.01 rad up to θ ) 0.500 rad.
resonance 2
position (eV)
a This work. Resonances 1 and 2 involve one and two negative eigenvalues (real part only) of the CSCF/ CMCSCF Hessian, respectively (see text). b See ref 36.
TABLE 3: 2P Be- Shape Resonance Positions and Widths from the Literature method static exchange phase shifta static exchange phase shift plus polarizability phase shifta static exchange cross sectionb static exchange plus polarizability cross sectionb ∆SCF with complex 14s16p Gaussian basis setc ∆SCF with complex 5s14p (Slater-type) basis setd singles, doubles, and triples complex CIe S matrix pole (XR)f second-order dilated electron propagator based on real SCFg
position width (eV) (eV) 0.77 0.20 1.20 0.16 0.70 0.76 0.32 0.10 0.57
1.61 0.28 2.60 0.14 0.51 1.11 0.30 0.15 0.99
0.62 0.61 0.48 0.54 0.54 0.53
1.00 1.00 0.82 0.82 0.78 0.85
biorthogonal dilated electron propagator (basis set 14s11p)h a) zeroth order b) quasiparticle second order c) second order d) quasiparticle third order e) OVGF third order f) third order f
Figure 2. The θ-trajectories for resonance 2 (see text) using CSCF, CMCSCF with a 2s2p CAS, and CMCSCF with a 2s2p3s3p CAS. The values of R (Ropt) are 1.020, 0.985, and 0.980, respectively. In each of the trajectories, θ started with a value of 0 at the top where Im(ε) ) 0 and incremented linearly in steps of 0.01 rad up to θ ) 0.550 rad.
shall report the total energy of the continuum Be- species relative to that of Be atom as
ε(η) ) EN+1 (η) - EN0 c
(24)
where EN+1 and EN0 are the total energies of the (N + 1) electron c Be- state under investigation and the N electron ground-state of the neutral Be atom, respectively, and the subscripts c and 0 refer to continuum and bound states, respectively. The imaginary part of ε(η) is plotted as a function of its real part for incremental θ values starting from θ ) 0 at a series of constant R values (θ trajectories). The θ trajectories, which show maximum stability for resonance 1 and resonance 2, are shown in Figures 1 and 2, respectively. In order to compare each of them with complex SCF, that is, SCF using a complex scaled Hamiltonian (CSCF), we also calculated the total energies using a single determinantal wave function using eqs 13-19. Resonances from CSCF were obtained for only one and two negative eigenvalues of the Hessian (resonances 1 and 2, respectively), similar to the CMCSCF cases. The resonance positions and widths determined from these
a See ref 40. b See ref 41. c See ref 15. d See ref 13. e See ref 17. See refs 42 and 43. g See ref 26. h See ref 25.
plots are listed in Table 2. The numbers from our previous calculations using the M1 method36 on the Be atom are listed in the same table for comparison. For resonance 1, the resonance positions are found to be 0.31, 0.30, and 0.31 eV for CSCF, CMCSCF with 2s2p CAS, and CMCSCF with 2s2p3s3p CAS, respectively, and the respective widths are found to be 0.40, 0.48, and 0.49 eV. For resonance 2, the positions are found to be 0.68, 0.71, and 0.73 eV, and the widths are 0.58, 1.56, and 1.58, respectively. The resonance positions in the current study for resonance 1 are ∼46% lower than those obtained in our previous study using the M1 method,36 on average, but for resonance 2, the positions are very close in the two sets of calculations. The widths from CMCSCF are ∼58% lower than those from the M1 method for resonance 1 and ∼40% higher for resonance 2, on average (except for CSCF, which is ∼60% lower than the latter). This result is not surprising since our previous calculations with MCSTEP have shown that for very accurate electron affinities and ionization potentials, additional blocks of the M matrix are necessary. These blocks include additional nondynamical and some dynamical correlation corrections and allow for more orbital relaxation upon electron addition and removal.5 We have listed a summary of results obtained by other workers in Table 3. The positions for resonance 1 in this work are very close to the numbers reported for singles, doubles, and triples complex CI,17 whereas the results for resonance 2 are close to the positions and widths obtained from static exchange
16218 J. Phys. Chem. B, Vol. 112, No. 50, 2008
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Figure 3. Plot of radial probability densities of the resonant p orbital as a function of the radial distance from the nucleus (r) for CSCF, CMCSCF with 2s2p CAS, and CMCSCF with 2s2p3s3p CAS for resonance 1. (The plots for the two CMCSCF cases overlap each other).
Figure 4. Plot of radial probability densities of the resonant p orbital as a function of the radial distance from the nucleus (r) for CSCF, CMCSCF with 2s2p CAS, and CMCSCF with 2s2p3s3p CAS for resonance 2. (The plots for the two CMCSCF cases overlap each other).
phase shift calculation.40 Our results for resonance 1 and resonance 2 using CSCF, CMCSCF, and the M1 method are close to other reported numbers. However, all of the results by others listed in Table 3 were for only one resonance that they obtained and not two that we obtained with the CSCF and CMCSCF methods and the M1 method36 (Table 2). In order to see how an orbital (say, φ˜ b), especially the resonant p orbital, obtained from the CSCF and CMCSCF calculations look, we plotted its radial probability density, Pb(r), which we define as36 2 Pb(r) dr ) N-1 b r dr
∫0π sin θ dθ ∫02π dφ |φ˜b(r, θ, φ)|2 (25)
with the choice of Nb ) 〈φ˜ b|φ˜ b〉 so that Pb(r) g 0 ∀r g 0. The plots of Pb(r) for the resonant p orbital as a function of r are shown in Figures 3 and 4 for resonances 1 and 2, respectively. In both cases, the shape of the plots for the CMCSCF with 2s2p and 2s2p3s3p CAS choices are almost identical, although they differ quite a bit from that for the CSCF.
In case of resonance 1, the 〈r2〉1/2 ) [∫∞0 r2Pb(r)dr]1/2 values for the resonant p are estimated to be 13.96, 23.30, and 23.29 bohrs for CSCF, CMCSCF with 2s2p CAS, and CMCSCF with 2s2p3s3p CAS, respectively. For resonance 2, the respective values are 17.97, 17.73, and 17.72 bohrs. To see how the resonant p orbitals compare with the orbitals in the CAS, we calculated 〈r2〉1/2 of the 2s and the 2p orbitals as well at the CMCSCF stationary points where the resonances were discovered. In the case of resonance 1, the 〈r2〉1/2 values of the 2s and the 2p orbitals are 3.00 and 3.22 bohrs for 2s2p CAS and 3.04 and 3.43 bohrs for 2s2p3s3p CAS, respectively. In case of resonance 2, the respective values are 3.07 and 3.75 bohrs for 2s2p CAS and 3.11 and 3.93 bohrs for 2s2p3s3p CAS. These numbers show that the resonant p orbital lies away from the region of the active orbitals. The 〈r2〉1/2 values for the 2s orbital at the CSCF stationary points at the positions of resonances 1 and 2 are 3.02 and 3.19 bohrs, respectively. The difference in shapes of both the resonance orbitals and small 〈r2〉1/2 in case of CSCF for resonance 1 compared to those for CMCSCF are attributable to there being no correlating 2p orbital with CSCF. However, this does not affect the resonance energies much although it does affect the resonance width in case of resonance 2. The 〈r2〉1/2 values for the resonant p orbital from this study, except for the case of resonance 1 using CSCF, are larger than those estimated in the M1 method (∼18.50 and ∼10.30 bohrs for resonances 1 and 2, respectively)36 since with CSCF and CMCSCF, the orbitals are actually the “relaxed” orbitals of Be-. In case of the M1 method, a CSCF/CMCSCF Be atom initial state was used where the orbitals are not “relaxed”. In order to better account for relaxation, all five blocks of the M matrix must be included.5 The shapes of the radial probability density plots for the resonance 2 in the current work (Figure 4) are similar to those in the M1 method,36 except for the fact that the current ones are more spread-out and further out in space due to orbital and state relaxation. However, the plots for resonance 1 are quite different from those seen in case of the M1 method. This is caused by the greater overlap with the valence electrons in the M1 method than that in the CSCF/CMCSCF cases because the resonant electron lies further out in the latter. The changes in the shapes of some of the resonant orbitals and 〈r2〉1/2 upon going from M1 to the direct CSCF/CMCSCF indicates that relaxation effects are important for these resonances. There are also changes in going from CSCF to CMCSCF, as shown in this work, which demonstrates that nondynamical correlation also plays an important role. 4. Summary and Conclusions We have presented a summary of the theory behind quadratically convergent complex multiconfigurational self-consistent field theory (CMCSCF) that uses a complex scaled Hamiltonian. The use of a complex scaled Hamiltonian enables one to search for the resonances that lie in the continuum part of the Hamiltonian using a bound-states method such as multiconfigurational self-consistent field theory (MCSCF). In order to control the walk on the complex energy hypersurface for a CMCSCF calculation far from convergence, a modified constrained optimization algorithm is used, as outlined in the Theory section. The continuum 2P Be- states are investigated in search for shape resonances. To our knowledge, this is the first time that CMCSCF is directly used to obtain resonances. The quadratic convergence of the CMCSCF method, coupled with the use of
Investigation of 2P Be- Shape Resonances the modified constraint algorithm, enables convergence with a tolerance of 1.0 × 10-10 au for the largest element of the energy gradient within 10 iterations or less, starting from a reasonable initial guess. Computational times necessary to obtain resonance positions and widths are modest. Using CSCF (complex SCF, i.e. CMCSCF in the limit of a single configuration) and CMCSCF, we have shown that there exist two shape resonances of 2P symmetry for Be-. This was also shown by our previous calculations using the M1 method on the Be atom.36 The resonance positions are fairly close to each other. Our best CMCSCF calculations (with the 2s2p3s3p CAS) for the resonance positions are 0.31 and 0.73 eV, and the respective widths are 0.49 and 1.58 eV for the two resonances. For the calculations by others listed in Table 3, apparently only one but not both of the two resonances present in this region was obtained. The radial probability density plots and the 〈r2〉1/2 values from this work on Be- shape resonances show that both relaxation and nondynamical correlation effects are important. This justifies the use of a MCSCF-based method to study resonances. In the future, the current work will be followed up with investigating shape, the Feshbach and the Auger resonances of other chemically and physically interesting atoms and molecules using CMCSCF. The current CMCSCF work reported here will further pioneer the development of complex multiconfigurational spin-tensor electron propagator (CMCSTEP) to study resonances by introducing complex scaling in all of the other blocks of the M matrix in addition to the first. Acknowledgment. We acknowledge the Robert A. Welch Foundation (Grant No. A-770) for the financial support and the National Science Foundation (Grant No. CHE-0541587). We also thank Dr. Dongxia Ma for her constructive comments and suggestions. We are grateful to Professor Manoj K. Mishra from the Indian Institute of Technology Bombay for initially providing us with valuable details on the method of complex scaling. References and Notes (1) Yeager, D. L.; Jørgensen, P. J. Chem. Phys. 1979, 71, 755. (2) Jørgensen,P.; Simons, J. Second Quantization-Based Methods in Quantum Chemistry; Academic Press: New York, 1981. (3) Olsen, J.; Yeager, D. L.; Jørgensen, P. AdV. Chem. Phys. 1983, 54, 1. (4) Graham, R. L.; Golab, J. T.; Yeager, D. L.; Olsen, J.; Jørgensen, P.; Harrison, R.; Zarrabian, S. J. Chem. Phys. 1986, 85, 6544. (5) Golab, J. T.; Yeager, D. L. J. Chem. Phys. 1987, 87, 2925.
J. Phys. Chem. B, Vol. 112, No. 50, 2008 16219 (6) Graham, R. L.; Yeager, D. L.; Rizzo, A. J. Chem. Phys. 1989, 91, 5451. (7) Rizzo, A.; Yeager, D. L. J. Chem. Phys. 1990, 93, 8011. (8) Aguilar, J.; Combes, J. Commun. Math. Phys. 1971, 22, 269. (9) Balslev, E.; Combes, J. Commun. Math. Phys. 1971, 22, 280. (10) Simon, B. Commun. Math. Phys. 1972, 27, 1. ¨ hrn, Y. J. Chem. Phys. 1983, 79, 5494. (11) Mishra, M.; Goscinski, O.; O (12) Simons, J. Annu. ReV. Phys. Chem. 1977, 28, 15. (13) Rescigno, T. N.; McCurdy, J. C. W.; Orel, A. E. Phys. ReV. A 1978, 17, 1931. (14) McCurdy, C. W.; Rescigno, T. N. Phys. ReV. Lett. 1978, 41, 1364. (15) McCurdy, C. W.; Rescigno, T. N.; Davidson, E. R.; Lauderdale, J. G. J. Chem. Phys. 1980, 73, 3268. (16) McCurdy, C. W.; Turner, J. L. J. Chem. Phys. 1983, 78, 6773. (17) McCurdy, C. W.; McNutt, J. F. Chem. Phys. Lett. 1983, 94, 306. (18) Rescigno, T.; McCurdy, C. W. Phys. ReV. A 1985, 31, 624. (19) Barrios, R.; Simons, J. J. Phys. Chem. B 2002, 106, 7991. ¨ hrn, Y.; Froelich, P. Phys. Lett. A 1981, 81, 339. (20) Mishra, M.; O (21) Mishra, M. K.; Medikari, M. N. AdV. Quantum Chem. 1996, 27, 223. (22) Venkatnathan, A.; Mahalakshmi, S.; Mishra, M. K. J. Chem. Phys. 2001, 114, 35. (23) Mishra, M. K.; Venkatnathan, A. Int. J. Quantum Chem. 2002, 90, 1334. (24) Sajeev, Y.; Mishra, M. K.; Vaval, N.; Pal, S. J. Chem. Phys. 2003, 120, 67. (25) Venkatnathan, A.; Mishra, M.; Jensen, H. J. A. Theor. Chim. Acta 2000, 104, 445. (26) Donnelly, R.; Simons, J. J. Chem. Phys. 1980, 73, 2858. (27) Moiseyev, N.; Hirshfelder, J. O. J. Chem. Phys. 1988, 88, 1063. (28) Hazi, A. U.; Taylor, H. S. Phys. ReV. A 1970, 1, 1109. (29) Liebman, J. F.; Yeager, D. L.; Simons, J. Chem. Phys. Lett. 1977, 48, 227. (30) Riss, U. V.; Meyer, H.-D. J. Phys. B 1993, 26, 4503. (31) Sommerfeld, T.; Riss, U. V.; Meyer, H.-D.; Cederbaum, L. S. Phys. ReV. A 1997, 55, 1903. (32) Santra, R.; Cederbaum, L. S. J. Chem. Phys. 2002, 117, 5511. (33) Feuerbacher, S.; Sommerfeld, T.; Santra, R.; Cederbaum, L. S. J. Chem. Phys. 2003, 118, 6188. (34) Yeager, D. L.; Mishra, M. K. Int. J. Quantum Chem. 2005, 104, 871. (35) Raimes, S. Many Electron Theory; Academic Press: London, 1973. (36) Samanta; K.; Yeager, D. L. Obtaining Positions and Widths of Scattering Resonances from a Complex Multiconfigurational Field State using the M1 Method. Int. J. Quantum Chem. submitted for publication. (37) Jørgensen, P.; Swansrøm, P.; Yeager, D. L. J. Chem. Phys. 1983, 78, 347. (38) Fletcher, R. Practical Methods of Optimization; Wiley: New York, 1980. (39) Moiseyev, N.; Friedland, S.; Certain, P. R. J. Chem. Phys. 1981, 74, 4739. ¨ hrn, Y. Phys. ReV. A 1979, 19, 43. (40) Kurtz, H. A.; O (41) Kurtz, H. A.; Jordan, K. J. Phys. B 1981, 14, 4361. (42) Krylstedt, P.; Rittby, M.; Elander, N.; Bra¨ndas, J. J. Phys. B 1987, 20, 1295. (43) Krylstedt, P.; Elander, N.; Bra¨ndas, J. J. Phys. B 1988, 21, 3969.
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