Athos Agathocleous

Ama Z Rohatiner

Simon Rule

Hannah Hunter

Jonathan Paul Kerr

Sally M Neeson

Janet Matthews

Sandra Strauss

Silvia Montoto

W M Johnson

Agathoclis Agathocleous

Peter Johnson

John Radford

Andrew Lister
email: a.lister@qmul.ac.uk

T Andrew Lister

WEEKLY VERSUS TWICE WEEKLY BORTEZOMIB GIVEN IN CONJUNCTION WITH RITUXIMAB IN PATIENTS WITH RECURRENT FOLLICULAR LYMPHOMA, MANTLE CELL LYMPHOMA AND WALDENSTRÖM MACROGLOBULINAEMIA

Keywords: Bortezomib, Rituximab, B-cell malignancies Default) Arial, Complex (Default) Arial, Complex bortezomib, rituximab, B-cell malignancies Bortezomib, Rituximab, B-cell malignancies Page 27: [2] Formatted

published or not. The documents may come

INTRODUCTION

The concept of 'targeted' therapy, first proposed by Erhlich (1890) came to fruition in the management of malignant lymphoma with the demonstration of single-agent efficacy for the monoclonal antibody, anti-CD20 (rituximab) in B-cell malignancy. Most effective, and licensed originally for use alone, in follicular lymphoma (FL) [START_REF] Mclaughlin | Rituximab chimeric anti-CD20 monoclonal antibody therapy for relapsed indolent lymphoma: half of patients respond to a four-dose treatment program[END_REF][START_REF] Hainsworth | Rituximab monoclonal antibody as initial systemic therapy for patients with lowgrade non-Hodgkin lymphoma[END_REF][START_REF] Hainsworth | Rituximab as first line and maintenance therapy for patients with indolent Non-Hodgkin's Lymphoma[END_REF][START_REF] Witzig | Rituximab Therapy for Patients With Newly Diagnosed, Advanced-Stage, Follicular Grade I Non-Hodgkin's Lymphoma: A Phase II Trial in the North Central Cancer Treatment Group[END_REF]), it also induces responses in mantle cell lymphoma (MCL) (Foran et al, 2000;[START_REF] Ghielmini | Effect of Single-Agent Rituximab Given at the Standard Schedule or As Prolonged Treatment in Patients With Mantle Cell Lymphoma: A Study of the Swiss Group for Clinical Cancer Research (SAKK)[END_REF][START_REF] Tobinai | Monoclonal Antibody Therapy for B-Cell Lymphoma: Clinical Trials of an Anti-CD20 Monoclonal Antibody for B-Cell Lymphoma in Japan[END_REF], and Waldenström macroglobulinaemia (WM) [START_REF] Treon | CD20-directed antibody-mediated immunotherapy induces responses and facilitates hematologic recovery in patients with Waldenstrom's macroglobulinemia[END_REF]Dimopoulos, et al 2002;[START_REF] Gertz | Multicenter phase 2 trial of rituximab for Waldenstrom macroglobulinemia (WM): An eastern Cooperative Oncology Group Study (E3A98)[END_REF][START_REF] Treon | Extended rituximab therapy in Waldenström's macroglobulinemia[END_REF]. The limited toxicity profile makes it an ideal partner for cytotoxic chemotherapy and 'chemo-immunotherapy' has now been shown to prolong not only freedom from recurrence but also survival in FL [START_REF] Marcus | CVP chemotherapy plus rituximab compared with CVP as firstline treatment for advanced follicular lymphoma[END_REF]Hiddeman et al, 2005;Kropff et al, 2005;van Oers et al, 2007;[START_REF] Herold | Frontline therapy with rituximab added to the combination of cyclophosphamide, doxorubicin, vincristine, and prednisone (CHOP) significantly improves the outcome for patients with advanced-stage follicular lymphoma compared with therapy with CHOP alone: results of a prospective randomized study of the German Low-Grade Lymphoma Study Group[END_REF]. Several studies in MCL have likewise shown a significant advantage for the addition of rituximab in terms of response rate, complete response rate and 'time to treatment failure' [START_REF] Howard | Rituximab and CHOP Induction Therapy for Newly Diagnosed Mantle-Cell Lymphoma: Molecular Complete Responses Are Not Predictive of Progression-Free Survival[END_REF][START_REF] Lenz | Immunochemotherapy With Rituximab and Cyclophosphamide, Doxorubicin, Vincristine, and Prednisone Significantly Improves Response and Time to Treatment Failure, But Not Long-Term Outcome in Patients With Previously Untreated Mantle Cell Lymphoma: Results of a Prospective Randomized Trial of the German[END_REF],but only one has thus far shown a survival advantage (Forstpointner et al, 2004).

The proteasome and in particular, the NFκB pathway is also a potentially attractive target in malignant lymphoma (de Martino et al, 1999& Adams et al, 1999). A phase I study of the proteasome inhibitor, bortezomib in patients with refractory haematological malignancy identified a feasible dosing schedule, the most debilitating clinical toxicities being fatigue and -4 -mostly reversible, but sometimes painful, peripheral neuropathy; thrombocytopenia was also observed (Orlowski et al, 2002) . Evidence of efficacy was most impressive in multiple myeloma but responses were also seen in FL, MCL and WM (Orlowski, et al, 2002).

In subsequent phase II studies, bortezomib was generally given twice weekly, at doses ranging from 1-1.6mg/m 2 . Activity was observed in patients with FL who had received multiple previous therapies (Di Bella et al, 2008;[START_REF] O'connor | Phase II Clinical Experience With the Novel Proteasome Inhibitor Bortezomib in Patients With Indolent Non-Hodgkin's Lymphoma and Mantle Cell Lymphoma[END_REF][START_REF] Strauss | Bortezomib Therapy in Patients With Relapsed or Refractory Lymphoma: Potential Correlation of In Vitro Sensitivity and Tumor Necrosis Factor Alpha Response With Clinical Activity[END_REF] and in MCL, [START_REF] O'connor | Phase II Clinical Experience With the Novel Proteasome Inhibitor Bortezomib in Patients With Indolent Non-Hodgkin's Lymphoma and Mantle Cell Lymphoma[END_REF][START_REF] Goy | Phase II Study of Proteasome Inhibitor Bortezomib in Relapsed or Refractory B-Cell Non-Hodgkin's Lymphoma[END_REF][START_REF] Strauss | Bortezomib Therapy in Patients With Relapsed or Refractory Lymphoma: Potential Correlation of In Vitro Sensitivity and Tumor Necrosis Factor Alpha Response With Clinical Activity[END_REF]. However, a subsequent multi-centre trial (in MCL) suggested a rather lower response rate (31%), with a median duration of response of 9 months in a heavily pre-treated population (Fisher et al, 2006). Again, the principal toxicities were gastro-intestinal tract symptoms, peripheral neuropathy and thrombocytopenia [START_REF] O'connor | Phase II Clinical Experience With the Novel Proteasome Inhibitor Bortezomib in Patients With Indolent Non-Hodgkin's Lymphoma and Mantle Cell Lymphoma[END_REF][START_REF] Strauss | Bortezomib Therapy in Patients With Relapsed or Refractory Lymphoma: Potential Correlation of In Vitro Sensitivity and Tumor Necrosis Factor Alpha Response With Clinical Activity[END_REF][START_REF] O'connor | Schedule of bortezomib administration may be an important determinant of single-agent activity in patients with relapsed or refractory follicular (FL) lymphoma and mantle cell lymphoma (MCL)[END_REF][START_REF] Goy | Phase II Study of Proteasome Inhibitor Bortezomib in Relapsed or Refractory B-Cell Non-Hodgkin's Lymphoma[END_REF]Fisher et al. 2006;Gerecitano et al, 2009& de Vos et al 2009).

Bortezomib has also been shown to be highly active in WM. On the basis of in vitro data, using cell lines and cells from patients with WM [START_REF] Mitsiades | Novel biologically based therapies for Waldenstrom's macroglobulinemia[END_REF][START_REF] Roccaro | Dual targeting of the proteasome regulates survival and homing in Waldenström macroglobulinemia[END_REF][START_REF] Leleu | Targeting NF-B in Waldenstrom macroglobulinemia[END_REF], the drug was first used alone, in patients with recurrent or refractory disease.

Unusually for such a patient population, 'major response' rates (Kimby et al, 2006) of >50% were observed [START_REF] Treon | Multicenter Clinical Trial of Bortezomib in Relapsed/Refractory Waldenstrom's Macroglobulinemia: Results of WMCTG Trial 03-248[END_REF]Chen et al, 2007;Dimopoulos et al, 2005[START_REF] Goy | Phase II Study of Proteasome Inhibitor Bortezomib in Relapsed or Refractory B-Cell Non-Hodgkin's Lymphoma[END_REF].

Formatted: Font: Italic Deleted: peripheral neuropathy.

Deleted: Evidence of efficacy was most impressive in multiple myeloma but responses were also seen in FL, MCL and WM (Orlowski, et al, 2002).

DESIGN AND METHODS

Study design

The protocol was approved by the Ethics Committees of the four participating hospitals; all patients gave informed consent. The study was carried out under the auspices of the NCRI Lymphoma Clinical Studies Group. This was originally conceived as a phase I/II study, the objective of the phase I component being to determine the optimal weekly or twiceweekly dose of bortezomib when given in conjunction with rituximab.

However, after the first 7 patients had been treated (in the Phase I component), in the light of data from a phase II study conducted in the United States, [START_REF] Leonard | Phase I/II Trial of Bortezomib + CHOP-Rituximab in Diffuse Large B Cell (DLBCL) and Mantle Cell Lymphoma (MCL): Phase I Results[END_REF] the protocol was amended to proceed directly to the randomised phase II component. The primary end points were: toxicity, which was monitored at each visit (using WHO criteria) and overall response rate. Secondary end points were freedom from progression and survival.

Deleted: In the latter and subsequent studies in other histological subtypes of lymphoma, thrombocytopenia and mostly reversible, but sometimes painful, peripheral neuropathy were identified as the major clinical problems associated with the drug (Orlowski et al, 2002;[START_REF] O'connor | Phase II Clinical Experience With the Novel Proteasome Inhibitor Bortezomib in Patients With Indolent Non-Hodgkin's Lymphoma and Mantle Cell Lymphoma[END_REF][START_REF] Strauss | Bortezomib Therapy in Patients With Relapsed or Refractory Lymphoma: Potential Correlation of In Vitro Sensitivity and Tumor Necrosis Factor Alpha Response With Clinical Activity[END_REF][START_REF] O'connor | Schedule of bortezomib administration may be an important determinant of single-agent activity in patients with relapsed or refractory follicular (FL) lymphoma and mantle cell lymphoma (MCL)[END_REF][START_REF] Goy | Phase II Study of Proteasome Inhibitor Bortezomib in Relapsed or Refractory B-Cell Non-Hodgkin's Lymphoma[END_REF]Fisher et al. 2006;Gerecitano et al, 2009& de Vos et al 2009) 1. Histological sub-types comprised: MCL: 21, FL: 17 and WM: 11. The median number of prior treatments was 2, range 1-7. Thirty-six of the 49 patients had bone marrow infiltration at the time of study entry; nine had previously received high-dose treatment supported by autologous peripheral blood progenitor cells.

Treatment

Phase I component: the first 7 patients were randomly allocated to receive bortezomib intra-venously (i/v), at the following doses: in Arm A: 0.7mg/m 2 (3 patients) and 1mg/m 2 (2 patients), twice weekly, or in arm B:

1.3mg/m 2 (2 patients),weekly.

Phase II component: Forty-two patients were randomised to receive one of two different doses and schedules of administration as shown in Figure I. Arm A comprised: bortezomib given at a dose of 1.3mg/m 2 i/v, on days 1,4,8 and 11, of a 21-day cycle, to a maximum of 8 cycles, rituximab :375mg/m 2 being given on day 1. In arm B, bortezomib was given at a dose of 1.6mg/m 2 on days 1,8,15 and 22 of a 35-day cycle, to a maximum of 6 cycles. In order for the total dose of rituximab to be equivalent, the antibody was given on the same days as bortezomib but only in cycles 1 and 4 (i.e. a total of 8 infusions in both arms of the study)

Formatted: Not Superscript/ Subscript Formatted: Superscript (Cheson et al,1999). Response in patients with WM was assessed using the Fourth International Workshop Criteria [START_REF] Kimby | Update on Recommendations for Assessing Response from the Third International Workshop on Waldenstrom's Macroglobulinemia[END_REF].

Progression-free survival is the interval between the date of assessment of response and evidence of progression. Overall survival is the interval from the time of commencing trial therapy until death or date of last follow-up.

RESULTS

Phase I Component:

In general, toxicity was minimal, however, one patient with bone marrow infiltration at the time of starting treatment was withdrawn after the first cycle due to prolonged thrombocytopenia and concurrent progression.

There was no evidence of response in the 6 other patients at the designated time of evaluation, although one patient with WM subsequently had a continuing decline in paraprotein from 26 to 10 g/L over a three year period, only having evidence of progression after a period of 4 years. Four of the 7 patients are alive, 2 having had further treatment. Cycles 2, + 3,5 and6 ¶ Bortezomib:1.6mg/m 2 i.v. on days 1, 8, 15 and22

Phase II Component:

Clinical Toxicity (Table 2)

There were no deaths directly related to the therapy, although two patients died of infection after the blood count had recovered. Planned therapy was however completed in only 14/42 patients. Twenty-eight were withdrawn for the following reasons: toxicity:16, progressive lymphoma: 7 and 'patient choice': 5. Four of the patients in the latter group asked to be withdrawn because they found the side effects unacceptable.

The number of patients withdrawn from each arm of the study because of toxicity and the specific reasons for cessation of treatment according to arm are shown in Table III; they were approximately equivalent. Overall, grade 1 or 2 neurotoxicity was observed in 43% of patients treated twice weekly and 48% of those receiving treatment weekly. Grade 3 or 4 neurotoxicity (in most cases, painful neuropathy) developed in 14% and 19% of patients in the 2 groups respectively. In all cases it was reversible. Three patients were withdrawn from the trial because of gastro-intestinal (g/i) tract toxicity.

Overall, grade I or 2 g/i toxicity was observed in 48%(nausea) and 52%

(diarrhoea) of patients receiving twice weekly treatment and 58% and 62% of patients receiving bortezomib weekly.

Haematological toxicity (Table 2),

Grade I or 2 toxicity occurred in more than one third of patients, being observed equally in the two arms of the study. There was a trend for Grade 3 or 4 haematological toxicity to be more severe in arm A, in which bortezomib was given twice weekly, but no significant difference.

Treatment was delayed in 14 patients, by 1-21 days, for various reasons but only in 2 cases because of neutropenia or thrombocytopenia.

Response (Table 3)

The

Progression-free survival

Ten of the 28 patients who responded remain well without progression between 1 and 3.5 years. Four patients with WM remain asymptomatic without further treatment for between 2 and 2.5 years, five have progressed, (progression being defined as a >25% increase in the M band).

Overall Survival (Fig. 2) [START_REF] Kahl | The VcR-CVAD Regimen Produces a High Complete Response Rate in Untreated Mantle Cell Lymphoma (MCL): First Analysis of E1405 -A Phase II Study of VcR-CVAD with Maintenance Rituximab for MCL[END_REF] and 'R-CHOP' [START_REF] Ruan | CHOP-R + Bortezomib as Initial Therapy for Mantle Cell Lymphoma (MCL)[END_REF] in newly-diagnosed patients with mantle cell lymphoma with very high response rates and CR rates of >70%.

Twenty
It is heartening for all three subtypes of lymphoma, that the results appear better than those reported in almost all studies for either drug given alone, in similar circumstances. Certainly, the combination warrants further evaluation. Particularly encouraging are the findings in the small number of patients with WM. Whilst no complete remissions were observed, partial responses were observed in 9/10 patients and 4

of the patients who responded continue well, transfusion-free without progression, between 2 and 2.5 years following completion of therapy.

Similar results have recently been reported in two studies from the Dana Farber Cancer Institute (Ghobrial, in press, 2010[START_REF] Treon | Primary Therapy of Waldenström Macroglobulinemia With Bortezomib, Dexamethasone, and Rituximab: WMCTG Clinical Trial 05-180[END_REF].

In the first of these, in newly diagnosed patients, dexamethasone was included and in an illness where previously, complete responses were virtually never seen, a small number of patients entered CR and the 'major response' rate was 83%. A phenomenon specific to WM is reflected in the most recent classification of response in this illness (Kimby et el, 2006). It is now recognised that responses to Bortezomib when given in conjunction with rituximab may occur late and may Even though largely reversible, as in myeloma (San Miguel et al, 2006), the neuropathy was distressing and debilitating for a significant proportion of patients, as was the case for similarly pre-treated patients with lymphoma in another report (de Vos et al, 2009). Very close monitoring and asking patients specifically about symptoms of 'early' neurotoxicity, with appropriate treatment modification is mandatory and may reduce the severity of the problem [START_REF] Richardson | Frequency, Characteristics, and Reversibility of Peripheral Neuropathy During Treatment of Advanced Multiple Myeloma With Bortezomib[END_REF]. Fisher, R.I., Bernstein, S.H., Kahl, B.S., Djulbegovic, B., Robertson, M.J., de Vos, S., Epner, E., Krishnan, A., Leonard, J.P., Lonial, S., Stadtmauer, E.A., O'Connor, O.A., Shi, H., Boral, A.L., & Goy. A. (2006) Multicenter ). Only 14Fourteen /42 patients completed therapy; 28 beingwere withdrawn (toxicity 16, progression 7, and 'patient choice' 5). The main clinical toxicities were neurologicalperipheral neuropathy and gastrointestinal. tract symptoms. Grade I/II haematological toxicity occurred in >33%. >1/3[A4]. The overall response rate was 28/42 (67%,i.e. 7), [A5]i.e.7 complete and 21 partial 21 partial responses). ,Fourteen of 14/21 patients in each arm responded; overall response rates by histology were: FL 8/15, : MCL 11/19,FL 8/15,and WM 9/10. [A6]Twelve of 14 patients who completed therapy responded, as did 16/28 withdrawnin whom the planned treatment was not .completeddid not. Ten of the /28 responding patients who responded remain progression-free( atfor between 1 -one and 1-3.5 years),, 22/42 are alive, (atbetween 1.4 -and -4 years). The main toxicities were peripheral neuropathy and gastro-intestinal tract symptoms. There was no difference in Ttoxicity andor efficacy were equivalent between the two schedules. Thise combination has significant toxicity is toxic[A7] but is effective, and warrants further investigation, particularly in patients with WM.

Formatted

Formatted

  On the basis of these preliminary results, a phase I/II randomised study of bortezomib given in conjunction with rituximab was initiated at four hospitals in the United Kingdom. The objective was to compare two different doses and schedules of administration of bortezomib in patients with recurrent and/or refractory FL, MCL and WM.

  of patients in the Phase II part of the study was based on the following considerations: assuming a 50% response rate to rituximab alone (from historical, control data), in order to demonstrate that the combination of bortezomib + rituximab was more effective than antibody alone, 20 patients were to be treated initially. If 16 responses were seen, a further 20 patients would be recruited. If between 7 and 15 responses were seen, a further 30 would be required. The planned interim analysis was duly conducted after 20 patients had been entered. On the basis of the safety and efficacy data, the decision was made to continue. Response was assessed at the end of 4 cycles in arm A and after 3 cycles in arm B (please see below) and one month after completion of therapy. Patients deemed to have 'stable disease' completed planned therapy (unless withdrawn because of subsequent progression, toxicity or 'patient choice'). For patients withdrawn from the study for any reason, formal assessment of response was undertaken at the time of that decision. Patients Fifty patients were originally recruited, one withdrew before starting treatment and has therefore been excluded from the analysis. Thus a total of 49 patients, 7 in the phase I component and 42 in the phase II of this report. Clinical characteristics at the time of study entry are shown in Table

  Same comment as in the summary "It might be more clear to say days 1, 4, 8, 11 of a 21-day cycle" Deleted: twice weekly, i/v, for two consecutive weeks Deleted: of each cycle.Deleted: on the first day of four consecutive were applied for patients with FL and MCL

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  Font: 12 pt, Complex Script Font: 12 pt Formatted: Line spacing: Double Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Complex Script Font: 12 pt Formatted: Font: 12 pt, Italic, Complex Script Font: 12 pt Formatted: Font: Italic Formatted: Line spacing: Double Formatted: Line spacing: Double Deleted: Fig. 1 ¶ ARM A (total -8 cycles) ¶ Bortezomib: 1.3mg/m 2 i.v. on days 1, 4, 8 & 11 of a 21-day cycle. ¶ Rituximab : 375mg/m 2 i.v. infusion on day 1 of each cycle ¶ ¶ ARM B (total -6 cycles) ¶ Cycles 1 and 4 ¶ Bortezomib:1.6mg/m 2 i.v. on days 1, 8, 15 and 22 of a 35-day cycle ¶ Rituximab : 375mg/m 2 i.v. infusion .( on same days as above) ¶

  , as aboveas above ¶ Cycle 4 Bortezomib:1.6mg/m 2 i.v. as above ¶ Rituximab: 375mg/m 2 i.v. inf. on same days as above ¶ Cycles 5 + 6 Bortezomib:1.6mg/m 2 i.v. as abovabove ¶ Definitions interim analysis of the results for the first 19 patients (one having been withdrawn prior to starting treatment), a further 23 were recruited.

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  Should clarify what this means-Where there were intolerable AE's, were the unsatisfied with the responses, etc? Deleted: .Deleted: allComment [A11]: What were the common manifestations of the G3-4 PN events?Comment [A12]: Did this result in dose reductions or delays?Comment [A13]: The "nausea and diarrhea…" statement is more of an observation for the discussion Deleted: Nausea and diarrhoea were also significant problems.

Formatted

  years, as was the case in one patient with WM treated in the Phase I component of the study.Against this background, the toxicity of the treatment is of paramount importance. Whilst meaningful responses were observed in the majority of patients withdrawn from the study, for whatever reason, it is disturbing that only 14/42 patients were able to complete the planned therapy. A quarter developed 'intolerable' side effects, the most serious being painful peripheral neuropathy, and treatment was therefore stopped, without prior dose reduction. The incidence and severity of neuropathy were similar in the two groups of patients. Equivalent degrees of neurotoxicity have likewise been observed in

Formatted

  . (1890) On immunity with special reference to cell life. Proceedings of the Royal Society. Foran, J.M., Rohatiner, A.Z.S., Cunningham, D., Popescu, R.A., Solal-Celigny, P., Ghielmini, M., Coiffier, B., Johnson, P.W.M., Gisselbrecht, C., Reyes, F., Radford, J.A., Bessell, E.M., Souleau, B., Benzohra, A., Lister, T. .Andrew. (2000) European Phase II Study of Rituximab (Chimeric Anti-CD20 Monoclonal Antibody) for Patients With Newly Diagnosed Mantle-Cell Lymphoma and Previously Treated Mantle-Cell Lymphoma, Immunocytoma, and Small B-Cell Lymphocytic Lymphoma Journal of Clinical Oncology, 18, 317.

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was identical in the two treatment groups, namely, 14/21 (67%) in each arm. There was no significant difference in CR rate.

		Comment [A14]: As above-what
		about neuropathy?
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	MCL: 11/19 (58%), FL: 8/15 (53%) and WM: 9/10 (90%). Twelve of the 14 patients who completed the planned therapy responded (3 CR, 9 PR), v i e as did 16 (4 CR, 12 PR) of the 28 patients who did not complete treatment (7 because of disease progression). Sixteen of the latter 28 w
	had been withdrawn because of toxicity, but 12/16 responded. (including
	6/8 withdrawn specifically because of neuropathy) . In addition, four out
	of five patients withdrawn because of 'patient choice' had a PR.
	-11 -

overall response rate was 28/42 (67%) with seven complete responses (CR) and 21 partial responses (PR) being observed. The response rate (CR + PR)

Survival curves according to treatment arm are shown in Fig.2 DISCUSSION

	Deleted: ¶
	¶
	¶
	¶
	Deleted:
	These results, evaluating the combination of bortezomib and rituximab in
	B-cell malignancies show that the weekly and twice-weekly schedules,

-two of the original 42 patients remain alive, between 1.4 and 4 years, 10 without progression. Twelve patients have developed recurrent lymphoma but 10 of them remain well, (8 without further treatment and 2 having gone on to receive a reduced intensity allograft). Twenty patients have died, 16 following progression of lymphoma (8 having had further treatment. Four patients died without evidence of progression, 2 of pneumonia (one with a history of COPD), one of secondary acute myeloblastic leukaemia and one as a consequence of previous therapy.

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We are still looking into this (patient not at Barts).

  designed to deliver the same total amount of both drugs yield the same results in terms of both toxicity and efficacy.

	F o r
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	Deleted:
	-12 -

Although outcome assessment was performed at somewhat different timepoints (the total duration of treatment being longer in Arm B), this is unlikely to confound the response rates since the discrepancy is only 6 weeks.

	specifically, recurrent follicular lymphoma (Fowler et al, 2009) have
		Deleted: both
	reported high response rates (79% and 86% respectively), with complete
	responses being observed in >50% of patients. Bortezomib has also
	recently been combined with 'CVAD'
		Formatted: Line spacing: Double
	The results are in keeping with data from most other studies in which
	bortezomib has been combined with conventional chemotherapy in other
	settings (Barr et al, 2009, Ribrag et al, 2009; Sehn et al, 2009 & Leonard
	et al, 2005). One report has however suggested that when combined with F o r
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	e	Comment [A15]: I think this term
	v i e w	Formatted: Font: (Default) Arial, sounds too casual for this article. Deleted: 'patient friendliness' of Deleted: the weekly schedule Complex Script Font: Arial
	Two recent Phase II studies using bortezomib in combination with
	bendamustine and rituximab, in patients with recurrent/refractory
	'indolent' and mantle cell lymphoma (Friedberg et al, 2009), and
	-13 -

rituximab, bortezomib is better tolerated and has greater efficacy when given weekly, to patients with recurrent or refractory follicular or marginalzone lymphoma, (de Vos et al, 2009). Conversely, it was concluded in a phase II study of bortezomib given alone, weekly, (Gerecitano et al, 2009) that whilst the latter schedule was less toxic than twice-weekly bortezomib, (O'Connor et al, 2005) the overall response rate was inferior, with no complete remissions in patients with recurrent FL and MCL.

Nonetheless, despite this, the considerably greater ease of giving treatment on a weekly basis provides a compelling argument for adopting this schedule in the future, certainly when the drug is being given in conjunction with rituximab.

Table II . Haematological and Clinical Toxicity (arm A vs. arm B)

 II

		Arm A -21 pts. Velcade: 1.3mg/m 2 x 2/week	Arm B -21 pts. Velcade:1.6mg/m 2 weekly	Deleted: Deleted:
		Grade 1 or 2 (% of pts.)	Grade 3 or 4 (% of pts.)	Grade 1 or 2 (% of pts.)	Grade 3 or 4 (% of pts.)	Formatted: Font: 12 pt, Bold Formatted Table Deleted: 21 pts.
	HB	38		0		29	5	Deleted: ¶ 1.3mg/m 2
	WCC	38		14		43	0	Deleted: 21 pts.
							Deleted:
	Platelets	43		29		52	10	Deleted: ¶ 1.6mg/m 2
	Neutrophils	33		24		29	14	Formatted: Font: 12 pt
	Lymphocytes 14 Nausea 48 Diarrhoea 52 Lethargy 43	38 5 10 5 F o r	58 58 62 38	24 0 0 10	Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Deleted: ¶ Deleted: ¶ Deleted: ¶ Deleted: ¶
	Neuropathy Fatigue	43 52		14 10	48 48 e P	19 5
					e r	Deleted: )
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					-21 -

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Table III . Toxicity necessitating withdrawal from trial (Arm A vs. B)

 III

	Page 21 of 38	British Journal of Haematology
	Toxicity	Arm A	Arm B	Total
	Neuropathy	5	3	8
	GI	2	1	3
	Haemorrhage	0	1	1
	Chest infection	0	2	2
	Cardiac Fatigue & Lethargy Total	0 1 8 F o r	1 0 8	1 1 16
		P
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		-22 -

Table IV . Response to treatment in Phase II component of trial

 IV

	Response	Arm A Velcade 1.3mg/m 2	Arm B Velcade 1.6mg/m 2	Total
		x 2/week	Weekly
	CR	3	4	7
	PR	11	10	21
	'Stable Disease'	2	4	6
	Progression	5	5	10
	Total	21	21	42
	ORR	67%	67%	67%

Table V . Response rate according to histological subtype in Phase II component of trial

 V

	Response	Mantle Cell	Follicular	Waldenström
		Lymphoma	Lymphoma	Macroglobulinaemia
	CR	3	4	0
	PR	8	4	9
	'Stable	0	4	1
	disease'
	Progression	8	3	0
	Total	19	15	10
	Fig.1.

Treatment in Phase II component of trial

  Adams, J., Palombella, V.J., Sausville, E.A., Johnson, J., Destree, A., Lazarus, D.D., Maas, J., Pien, C.S., Prakash, S., Elliott, P.J. (1999) Proteasome Inhibitors: A Novel Class ofPotent and Effective Antitumor Agents. Cancer Research, 59, 2615-2622. Barr, P.M., Fu,P., Lazarus, H.M., Horvath, N., Gerson, S.L., Koc, O.N., Bahlis, N.J., Snell, M.R., Dowlati, A., Cooper, B.W. (2009) Phase I trial of fludarabine, bortezomib and rituximab for relapsed and refractory indolent and mantle cell non-Hodgkin lymphoma. British Journal of Haematology, 147, 89-96. Chen, C.I., Kouroukis, C.T., White, D., Voralia, M., Stadtmauer, E., Stewart, A.K., Wright, J.J., Powers, J., Walsh, W., & Eisenhauer, E.(2007) Bortezomib Is Active in Patients With Untreated or Relapsed Waldenstrom's Macroglobulinemia: A Phase II Study of the National Cancer Institute of Canada Clinical Trials Group. Journal of Clinical Oncology 25, 1570-1575. Cheson, B.D., Horning, S.J., Coiffier, B., Shipp, M.A., Fisher, R.I., Connors, JM., Lister, T.A., Vose, J., Grillo-López, A., Hagenbeek, A., Cabanillas, F., Klippensten, D., Hiddemann, W., Castellino, R., Harris, N.L., Armitage, J.O., Carter, W., Hoppe, R., & Canellos, G.P. (1999) Report of an International Workshop to Standardize Response Criteria for Non-Hodgkin's Lymphomas. American Society for Clinical Oncology, 17, 1244-1253.Di Bella, N., Taetle, R., Kolibaba, K.S., Boyd, T.E., Raju, R.N., Barrera, D.N., Cochran, E., Dien, P., Hakimian, D., Lyons, R.M., Schlegel, P.J, Vukelja, S.J., Boston, J., Boehm, K.A., Wang, Y., Asmar, L.(2010) Results of a phase 2 study of bortezomib in patients with relapsed or refractory indolent lymphoma. Blood 115, 475-480.

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	R e v i e w R e v i e w De Martino, G.N. & Slaughter, C.A. (1999) The Proteasome, a Novel R Protease Regulated by Multiple Mechanisms. The Journal of Biological e Chemistry, 274, 22123-22126. v Dimopoulos, M.A., Zervas, C., Zomas, A., Kiamouris, C., Viniou, N.A., i Grigoraki, V., Karkantaris, C., Mitsouli, C., Gika, D., Christakis, J., e Anagnostopoulos, N.. (2002) Treatment of Waldenstrom's Macroglobulinemia With Rituximab. Journal of Clinical Oncology, 20, 2327-2333. w	Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Complex Script Font: Arial Formatted: Font: Not Italic, Complex Script Font: Arial Formatted: Complex Script Font: Arial Formatted: Font: Not Italic, Complex Script Font: Arial Formatted: Complex Script Font: Arial
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de Vos, S., Goy, A., Dakhil, S.R., Saleh, M.N., McLaughlin, P., Belt, R., Flowers, C.R., Knapp, M., Hart, L., Patel-Donnelly, D., Glenn, M., Gregory, S.A., Holladay, C., Zhang, T., & Boral A.L. (2009) Multicenter Randomized Phase II Study of Weekly or Twice-Weekly Bortezomib Plus Rituximab in Patients With Relapsed or Refractory Follicular or Marginal-Zone B-Cell Lymphoma. Journal of Clinical Oncology, 27, 5023-5030.

  Phase II Study of Bortezomib in Patients With Relapsed or Refractory Mantle Cell Lymphoma. Journal of Clinical Oncology, 24, 4867-4874.

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	H., Wandt, H., Unterhalt, M., & Hiddemann W. (2004) The addition of	Formatted: Normal, Border:
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  Orlowski, R.Z., Stinchcombe, T.E., Mitchell, B.S., Shea, T.C., Baldwin, A.S., Stahl, S., Adams, J., Esseltine, D., Elliott, P.J., Pien, C.S., Guerciolini, R., Anderson, J.K., Depcik-Smith, N.D., Bhagat, R., Lehman, M.J.,Novick, S.C., O'Connor, O.A., & Soignet, S.L. (2002) Phase I Trial of the Proteasome Inhibitor PS-341 in Patients With Refractory Hematologic Malignancies. Journal of Clinical Oncology, 20, 4420-4427.The proteasome inhibitor, Bortezomib, is active in follicular lymphoma (FL), mantle cell lymphoma (MCL) and Waldenström macroglobulinaemia (WM). Standard treatment for FL and MCL now includes rituximab. The combination was evaluated in a Phase I and subsequently a , randomised Phase II study. In the latter, 42 fForty-two patients with recurrent/refractory FL, MCL(19),FL(13), or WM [A1](r10) received either: Bortezomib 1.3mg/m 2 on days 1,4,8 and 11 of a 21-day cyclex2/week, for two consecutive weeks of a 21-day cycle[A2] (maximum 8 cycles), with Rituximab 375mg/m 2 on day 1 (21 patients) or: Bortezomib 1.6mg/m 2 and Rituximab on days 1, 8,15 and 22 weekly for 4 weeks of a 35-day5-week cycle (maximum 6, Rituximab being given only in cycles 1 and 4[A3]

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British Journal of Haematology

Acknowledgements

This study was supported by a grant from Millenium Pharmaceuticals, who also kindly provided bortezomib for the study free of charge. We thank Hoffman-la-Roche for likewise providing Rituximab. We thank the referring physicians and the medical and nursing staff at the four hospitals for their expert care, Lynda Harrison and Nafti Lafon for help with data collection, and Elizabeth Hyde for preparing the manuscript.

Authors' Contributions

TAL, AR, SR, PJ, and JR are the principal physicians, and take primary responsibility for the paper. AR and TAL wrote the paper. TAL, AR, SR, PJ, JR, AA, HH, SN, SS, and SM were involved in the care of the patients. AA and TAL compiled and analysed the data. All authors reviewed the manuscript.

Conflicts of interest

JR has participated in


K Doll
email: k.doll@fkf.mpg.de

Analytical stress tensor and pressure calculations with the CRYSTAL code

Keywords: analytical gradient, stress tensor, pressure, enthalpy analytical gradient, stress tensor, pressure, enthalpy

The calculation of the stress tensor and related properties and its implementation in the CRYSTAL code are described. The stress tensor is obtained from the earlier implemented analytical gradients with respect to the cell parameters. Subsequently, the pressure and enthalpy is computed, and a test concerning the pressure driven phase transition in KI is used as an illustration. Finally, the possibility of applying external pressure is implemented. The constant pressure optimization offers an alternative optimization method, in addition to the already implemented optimization at constant volume.

Introduction

The calculation of the total energy has become an important part of electronic structure theory, and a set of methods and codes are available. Nowadays, density functional calculations are feasible for large systems; and in many cases such calculations have become routine work. After the total energy is computed, then, from the minimum of the total energy, the optimal geometry is obtained. This is usually facilitated with the availability of analytical gradients, see e.g. [1][START_REF] Pulay | Modern electronic structure theory[END_REF][START_REF] Schlegel | Modern electronic structure theory[END_REF].

After the equilibrium structure of a solid is determined, there are further properties of interest which are closely related to the total energy, such as the bulk modulus or elastic constants. Even more important is the situation when the energy as a function of the volume is known, and the pressure has to be determined. When the pressure has been computed, the enthalpy can be obtained, and the search for possible phase transitions can be performed. If the derivative of the energy with respect to the unit cell parameters is known, then from these derivatives, the aforementioned properties (pressure, enthalpy) can be computed. The stress tensor [START_REF] Nye | Physical Properties of Crystals[END_REF][START_REF] Landau | Lehrbuch der Theoretischen Physik, Band[END_REF] is closely related to the derivatives of the total energy with respect to the cell parameters. The ab initio calculation of the stress tensor was pioneered some time ago [START_REF] Nielsen | [END_REF]. In this article, the implementation of the stress tensor in the electronic structure code CRYSTAL [START_REF] Pisani | Hartree-Fock Ab Initio Treatment of Crystalline Systems[END_REF][START_REF] Dovesi | CRYSTAL2006[END_REF] is reported. The article is organized as follows: in section 2, the variables are defined. The definition of the stress tensor, tests concerning the accuracy of the implementation, the calculation of pressure and enthalpy, and some examples are given in section 3. In section 4, the possibility of applying external pressure is demonstrated; and the optimization at constant volume is compared with the optimization at constant pressure. Finally, the paper is summarized.

Variables

An elastic deformation is given by the expression:

r ′ = (1 + ǫ) r (1)
or

r ′ j = 3 k=1 (δ jk + ǫ jk )r k (2)
where δ jk is the Kronecker delta. Here, the case of pure strain is assumed, and then ǫ jk is the symmetric strain tensor [START_REF] Nye | Physical Properties of Crystals[END_REF]:

  ǫ 11 ǫ 12 ǫ 13 ǫ 12 ǫ 22 ǫ 23 ǫ 13 ǫ 23 ǫ 33   =   ǫ 1 1 2 ǫ 6 1 2 ǫ 5 1 2 ǫ 6 ǫ 2 1 2 ǫ 4 1 2 ǫ 5 1 2 ǫ 4 ǫ 3   (3)
The primitive cell is given by three vectors: a 1 , a 2 and a 3 . a ij are defined in such a way that a 11 = a 1x is the x-component of a 1 , a 12 = a 1y the y-component of a 1 , and a 13 = a 1z the z-component of a 1 , i.e.:

  a 1 a 2 a 3   =   a 1x a 1y a 1z a 2x a 2y a 2z a 3x a 3y a 3z   =   a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33   (4)
When a distortion is applied, then the cell parameters transform as:

a ′ ij = 3 k=1 (δ jk + ǫ jk )a ik (5)
The derivatives (see also [START_REF] Kudin | [END_REF]10])

F
∂a ′ ij ∂ǫ lm ǫlm=0 = lim ǫlm→0 a ′ ij (ǫ lm ) -a ′ ij (ǫ lm = 0) ǫ lm = 3 k=1 δ jl δ km a ik = δ jl a im (6)
and

∂E ∂ǫ lm ǫlm=0 = (7) lim ǫlm→0 E( a 1 ′ (ǫ lm ), a 2 ′ (ǫ lm ), a 3 ′ (ǫ lm )) -E( a 1 ′ (ǫ lm = 0), a 2 ′ (ǫ lm = 0), a 3 ′ (ǫ lm = 0)) ǫ lm = 3 i,j=1 ∂E ∂a ′ ij ∂a ′ ij ∂ǫ lm ǫlm=0 = 3 i,j=1 ∂E ∂a ij δ jl a im = 3 i=1 ∂E ∂a il a im
are required for the calculation of the stress tensor.

The stress tensor and related properties

The stress tensor is a symmetric tensor, and following the notation of [START_REF] Nye | Physical Properties of Crystals[END_REF] it is given as:

  σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33   =   σ 1 σ 6 σ 5 σ 6 σ 2 σ 4 σ 5 σ 4 σ 3   (8)
It can be computed from the total energy as

σ ij = 1 V ∂E ∂ǫ ij (9)
or equivalently, using equation 3:

σ i = 1 V ∂E ∂ǫ i (10)
where V = a 1 ( a 2 × a 3 ) is the volume of the unit cell. Here, the sign is according to the convention used in [START_REF] Nye | Physical Properties of Crystals[END_REF][START_REF] Landau | Lehrbuch der Theoretischen Physik, Band[END_REF].

In the case of hydrostatic pressure, the stress tensor becomes diagonal, and the pressure is obtained as

σ ij = -pδ ij (11)
The calculation of the stress tensor thus makes the pressure p and the enthalpy H = E + pV accessible. The implementation is straightforward: The derivatives -0.00092248 -0.00092197 σ 4 0 0 σ 5 -0.00000555 -0.00000564 σ 6 0 0 3 dimensional periodicity [11] and for the case of 1 and 2 dimensional periodicity [12]. In addition, this offers the possibility to apply external pressure: by adding a term -pδ ij to the stress tensor, an optimization at constant pressure p is implemented.

Test 1: Oxalic acid dihydrate

A first test was done with oxalic acid dihydrate, which has a monoclinic lattice, and the stress tensor is non-diagonal and thus not too trivial. The experimental geometry as determined in [13] was used, and basis sets as in [14] were employed. The stress tensor was determined numerically, by applying two elastic distortions with ǫ i = 0.001 and ǫ i = -0.001, and then performing a numerical differentiation:

σ numerical i = 1 V ∆E ∆ǫ i (12)
The energy was evaluated at the Hartree-Fock level, and this test therefore serves essentially to verify that the transformation in equation 7 is implemented properly, and to check the accuracy. The results are compared in table 1, and it is apparent, that a good accuracy is obtained.

Test 2:

The NaCl → CsCl phase transition in KI As a practical example, the phase transition from the NaCl to the CsCl structure in KI is studied. The earlier suggested procedure [START_REF] Dovesi | Quantum-Mechanical Ab-initio Calculation of the Properties of Crystalline Materials[END_REF] is to compute the energy as a function of the volume (E(V ) curve), then perform a fit for the function E(V ), obtain the pressure p as the derivative of the fitting function p = -∂E ∂V and then the enthalpy H = E + pV . With the present implementation, the pressure is readily available via the stress tensor. Thus this procedure can be simplified, and the pressure and enthalpy can be computed analytically. This requires that the stress tensor is of the form -pδ ij . In the present implementation, this was assumed to be fulfilled, if all non-diagonal components of the stress tensor are below a threshold value, and the diagonal elements do not differ by more than a threshold value.

The total energies of KI in the NaCl and CsCl structure were computed, at the level of the local density approximation, with basis sets as were used for KI in [16]. The E(V ) curves were obtained by fitting the computed energy values with a polynomial of degree four. This is displayed in figure 1 (top). Subsequently, the pressure as a function of the volume was computed analytically and numerically (figure 1, middle). When the analytical differentiation with respect to the volume (crosses and circles in figure 1) is compared with the numerical differentiation of the polynomials which fit E(V ) (full and dashed lines in figure 1), then the results are virtually identical. This confirms that the implementation is reasonable. Also, the minimum of the energy and the zero of the pressure agree well.

By adding pV to the energy, the enthalpy is obtained (figure 1, bottom). The analytically computed enthalpies (crosses and circles) were obtained using the analytical derivative for the pressure and subsequently computing the enthalpy, the numerically computed enthalpies (full and dashed lines) were obtained with the numerically computed pressure. Again, when comparing the results employing analytical or numerical differentiation of the E(V ) curve, then the enthalpy curves are virtually identical.

As a by product of this test of the implementation and accuracy, the critical pressure at the NaCl → CsCl phase transition is obtained from the intersection of the two enthalpies for the NaCl and the CsCl structure. This pressure is 1.0 GPa (present work), and in reasonable agreement with the experiment (1.9 GPa [17]) or earlier calculations (2.75 GPa [18]).

The fact that the stress tensor and the pressure is now available as an analytical derivative simplifies the calculation of further elastic properties such as the elastic constants. For example, the bulk modulus can now be obtained by one numerical differentiation of the pressure with respect to the volume (first part of equation 13), instead of two numerical differentiations (second part of equation 13) which was necessary earlier; a similar argument holds for the elastic constants.

B = -V ∂p ∂V = V ∂ 2 E ∂V 2 (13)

Applying external pressure

Once the stress tensor has been computed, it is subsequently one possible option to add external pressure. For this purpose, an external part σ external ij is added to the stress tensor as defined in equation 9: (14) and thus the total stress σ total ij is obtained. When this is zero, then this means that a system is described, where an external stress is applied. As the optimizer works with the gradient with respect to the cell parameter, the stress tensor σ total ij has to be back transformed to obtain the modified gradients with respect to the cell parameter. Thus, the matrix a ij of the cell vectors has to be inverted, with (a -1 ) jk being the inverse:

σ total ij = σ ij + σ external ij
3 j=1 a ij (a -1 ) jk = δ ik (15)
Then, using this in equation 7, the back transformation is obtained: a mn σ lm (16) Exploiting equation 16, we thus obtain for the total gradient with respect to the cell parameter, after having added the external stress:

∂E ∂a ij total = ∂E ∂a ij + V 3 m=1 a mi σ external jm ( 17
)
∂E ∂aij is thus the gradient with respect to the cell parameter without external stress, and ∂E ∂aij total the gradient with respect to the cell parameter in the presence of an external stress.

In the present implementation, the option of a general external stress tensor is possible. As the stress tensor must be symmetric, this means six input parameters. In principle any form is possible, with special cases such as hydrostatic or uniaxial pressure. The most important case is that of hydrostatic pressure, where the tensor is diagonal (see equation 11).

As a test, in figure 2, the E(V ) curve for Al 2 O 3 is displayed. The pluses are data points obtained with the present implementation where the energy is optimized in the presence of external (hydrostatic) pressure. The squares are data points computed with the energy optimization under the constraint of a fixed volume, as implemented in the CRYSTAL06 release [START_REF] Dovesi | CRYSTAL2006[END_REF]. As can be seen, the data from both methods agrees very well, and the data points lie well on the E(V ) curve, which was obtained with a polynomial fit of degree four.

Similarly, in figure 3, the E(V ) curve for oxalic acid dihydrate is displayed. This system is monoclinic and the stress tensor more general (in the case of nonhydrostatic pressure, the stress tensor is not diagonal). Again, the data from fixed volume optimization (squares) and the data from the constant pressure optimization (pluses) agree well and lie on a smooth curve. This confirms that the present implementation is reliable.

When comparing the number of steps required for the geometry optimization, it turned out that both schemes (constant pressure versus constant volume) require a similar number of steps. The most important parameter is to have a good initial guess.

Conclusion

In this article, the calculation of the stress tensor and the implementation in the electronic structure code CRYSTAL is described. The stress tensor is obtained from the analytical gradient with respect to the cell parameters, and subsequently pressure and enthalpy can be computed, all based on analytical expressions without the need to compute any numerical derivatives. An additional feature is to add terms to the stress tensor, which allows to apply external pressure. This implementation also facilitates the calculation of properties such as the bulk modulus or elastic constants, as only one more numerical derivative is required.

  ∂E ∂aij constitute the difficult part and were implemented in the past for the case of

Figure 1 .

 1 Figure1. The total energy as a function of the volume (top figure), the pressure as a function of the volume (middle figure), and the enthalpy as a function of the pressure (bottom). The crosses and circles refer to the points where the total energy was calculated and subsequently the pressure and enthalpy derived analytically. The full and dashed lines are obtained by fitting the energies as a function of the volume with a polynomial, and subsequently computing the derivative of this polynomial.

Figure 2 .

 2 Figure 2. The total energy, on the B3LYP level, as a function of the volume for Al 2 O 3 . The pluses are data points computed with the present implementation of applying external pressure. The squares are data points obtained from an optimization at constant volume. The full line is a fit through the data points.

Figure 3 .

 3 Figure3. The total energy, on the Hartree-Fock level, as a function of the volume for oxalic acid dihydrate. The pluses are data points computed with the present implementation of applying external pressure. The squares are data points obtained from an optimization at constant volume. The full line is a fit through the data points.

Table 1 .

 1 Stress tensor σij , in hartree/bohr 3 , computed for oxalic acid dihydrate.

	component	numerical	analytical
	σ 1	-0.00093087 -0.00093087
	σ 2	-0.00064440 -0.00064415
	σ 3

URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

Taylor & Francis and I.T. Consultant


Reinhard Schinke
email: rschink@gwdg.de

Mikhail V Ivanov
email: mivanov@gmx.de

Vibrational energy transfer in Ar-O_3 collisions: Comparison of rotational sudden, breathing sphere, and classical calculations

Keywords: breathing sphere, and classical calculations ozone, energy transfer, stabilization

published or not. The documents may come

I. INTRODUCTION

The recombination of ozone shows an intriguing isotope effect 1 which, in spite of numerous theoretical efforts, 2 is not yet fully understood. The rate coefficient for the formation of nonsymmetric isotopomers -for example 688 or 866, where 6 and 8 stand for 16 O and 18 O, respectively -depends linearly on the difference of zero-point energies of the possible O 2 dissociation products (∆ ZPE ). 3 This dependence has been explained by statistical, 4 classical trajectory, 5 and quantum mechanical 6 calculations. According to the experimental data 3 the rate coefficients for forming symmetric ozone isotopomers with ∆ ZPE = 0 -for example 666 or 686 -do not follow the linear dependence: They are all by 15%-20% smaller than expected from the results of the non-symmetric isotopomers. This apparent symmetry effect 4,7,8 where M represents a buffer gas. The density of vibrational quantum states in asymmetric ozone is roughly twice the density of symmetric or anti-symmetric vibrational states in symmetric ozone and this difference may influence the intramolecular dynamics of ozone as well as the energy exchange with M.

In a recent study 9 (hereafter termed I) we investigated the vibrational energy transfer in collisions of O 3 with Ar using a most simple quantum mechanical approximation: the breathing sphere approximation (BSA). Within the BSA the six-dimensional Ar -O 3 potential energy surface (PES) is first averaged over the two Ar -O 3 orientation angles θ and ϕ. This completely eliminates rotational transitions and thus drastically simplifies the scattering calculations. An inherent feature of the BSA is that for symmetric isotopomers transitions between symmetric and anti-symmetric ozone states are strictly forbidden. The main goal in I was to test whether the collisional energy transfer, and therefore the stabilization rate, is different for symmetric and non-symmetric molecules.

The main results of I can be summarized as follows: (1) The individual cross sections for vibrationally inelastic transitions n → n ′ show large fluctuations as function of n ′ ; the fluctuations reflect the nodal structures of the underlying vibrational wave functions of O 3 .

(2) The inelastic probabilities are sufficiently small that first-order perturbation theory can be used for analysis, even for initial states close to the dissociation threshold. (3) The energy transfer ∆E n is a strongly fluctuating function of the initial state n and therefore substantial averaging is mandatory for comparing the results for different isotopomers. (4) The BSA calculations show a strong symmetry effect for low initial ozone states, where the energy transfer for the non-symmetric molecules is by an order of magnitude larger than for the symmetric ones. However, for higher initial states with energies E n exceeding half of the dissociation energy, ∆E n is almost indistinguishable for symmetric and nonsymmetric molecules. (5) The results can be rationalized in terms of the polyad structure of the vibrational spectrum and several near-resonance conditions. ( 6) Except for the nonsymmetric isotopomers in the low-energy regime, ∆E n increases very strongly with E n and is proportional to (E n -E 0 ) k with k = 5-7. The main conclusion of I was that, within the BSA, the energy exchange between highly excited ozone complexes close to the dissociation threshold with the buffer gas is independent of the symmetry.

The BSA is a very crude approximation and is not expected to be quantitatively correct. In the present study we investigate the vibrational energy transfer in Ar -O 3 collisions by means of the more accurate infinite order sudden approximation (IOSA) for the rotational degrees of freedom. [11][12][13] Vibrationally inelastic scattering calculations are performed for fixed Ar -O 3 orientations (θ, ϕ) and the cross sections -rather than the interaction potential as in the BSA -are finally averaged over all orientations; the anisotropy of the six-dimensional PES is fully retained in the scattering calculations. 14 Averaging of the cross sections makes the IOSA calculations about two orders of magnitude more time-consuming than the BSA calculations. However, unlike in the BSA, transitions between symmetric and anti-symmetric states for symmetric molecules are allowed. The IOSA has been applied before for studying ozone recombination, 15,16 albeit in strongly limited ways.

II. CALCULATIONS

A. Quantum mechanical calculations

Calculations have been performed for two symmetric isotopomers, 666 and 686, and one non-symmetric one, 668. The corresponding vibrational wave functions Φ n for these three (non-rotating) isotopomers, required for expanding the scattering wave function, have been calculated by means of the filter diagonalization (FD) method. 17 The same simplified PES, V mod , used in I, has been employed in these bound state calculations. It has been obtained by removing the three long-range van der Waals wells by exponential extrapolation along the dissociation coordinate from a value inside the main well to the correct asymptotic limit. In addition, two of the three main ozone potential wells have been eliminated by extrapolation along the bond angle α from α = 85 • to 0. The extrapolations have been made so that V mod and its first derivatives are continuous. Details are described in I. The vibrational wave functions Φ n are functions of the internal ozone coordinates: R, the distance of the central O atom to the center of mass of the two end O atoms; r, the distance between the two end atoms; and γ, the angle between R and r. (R, r, γ) form a set of Jacobi coordinates.

The wave functions for symmetric isotopomers are either symmetric or anti-symmetric with respect to γ = π/2. All vibrational states up to the dissociation threshold -244, 254, and 259 for 666, 668, and 686, respectively --are considered in the scattering calculations. In an attempt to model in a simple way the O + O 2 continuum we additionally calculated, by the FD method, wave functions of states with energies above the dissociation limit. An absorbing potential at the grid boundary was not used in these calculations, i.e., these realvalued so-called "box states" are zero at the grid boundary. They represent artificial bound states and extend the true bound states, to which they are orthogonal, into the continuum.

The "box states" were calculated only for 666 (80 states) and 668 (90 states). They were not part of the main study, but included only in some test calculations.

The collision of Ar with O 3 is described by the polar coordinates ρ, θ, and ϕ defined with respect to a coordinate system, in which the center-of-mass of O 3 is the origin and O 3 lies in the XY plane with the central O atom located on the positive X axis. 18 According to this choice of coordinates, θ = 0 represents Ar approaching perpendicular to the plane of O 3 (i.e., along the Z axis) and θ = 90 • corresponds to in-plane collisions. For an in-plane collision, ϕ = 0 corresponds to Ar approaching along the X axis. The interaction potential V Ar-O 3 between the Ar atom and O 3 is written as a sum of three Ar-O potentials

V Ar-O (R i ),
where R 1 , R 2 , and R 3 are the three Ar -O distances. 9 The evaluation of V Ar-O 3 requires the determination of the R i for a given set of scattering coordinates (R, r, γ, ρ, θ, ϕ). This is done using the condition that the angular momentum vector of ozone is zero. 18 In the IOSA one has to solve an Ar -O 3 scattering problem for a set of fixed angles (θ, ϕ). The time-independent Schrödinger equation for the lth partial wave Ψ l is given by

- 2 2µ ∂ 2 ∂ρ 2 + 2 2µ l(l + 1) ρ 2 + ĥO 3 (R, r, γ) + V Ar-O 3 (R, r, γ, ρ, θ, ϕ) -E Ψ l = 0 , (1)
where the angles θ and ϕ enter only as parameters. Expanding Ψ l in terms of the eigenfunctions Φ n (R, r, γ) of ĥO 3 , one obtains the set of coupled equations,

d 2 dρ 2 - l(l + 1) ρ 2 + k 2 n ′ χ n ′ l (ρ|θ, ϕ) = 2µ 2 n ′′ V n ′ n ′′ (ρ|θ, ϕ)χ n ′′ l (ρ|θ, ϕ) (2)
for the angle dependent radial expansion functions χ n ′ l (ρ|θ, ϕ); k 2 n ′ = 2µ(E -E n ′ )/ 2 and E n ′ and µ are the vibrational energies of ozone and the Ar -O 3 reduced mass, respectively.

The potential matrix elements are defined by [START_REF] Unfortunately | the correct volume elements have been neglected by mistake in the integrals in Eqs[END_REF] V n ′ n ′′ (ρ|θ, ϕ) = dR dr dγ sin γ Φ n ′ V (R, r, γ, ρ, θ, ϕ)Φ n ′′ .

(

) 3
The Ar-O 3 interaction potential is strongly anisotropic as illustrated in Fig. 1, which shows a selection of diagonal matrix elements V nn (ρ) for 686 for several pairs (θ, ϕ). The quantum in Fig. 1). The nondiagonal coupling elements are also strongly angle dependent.

Solution of the coupled equations with the appropriate boundary conditions and summation over l yields vibrationally inelastic cross sections σ n ′ n (θ, ϕ). 9 All quantum mechanical calculations presented below are for a collision energy of E c = 200 cm -1 . The strong angle dependence of the V n ′ n implies a large anisotropy of the σ n ′ n (θ, ϕ) illustrated in Fig. 2 for the sum of inelastic cross sections. The cross sections for the symmetric isotopomers are symmetric with respect to ϕ = 180 • , while this symmetry does not exist for non-symmetric molecules. For θ = 0 the cross sections are independent of ϕ. The sum of inelastic cross sections for the same initial state varies by more than two orders of magnitude over the full sphere. The inelasticity is generally weakest for the perpendicular approach.

The final cross sections are obtained by averaging over the angles, 19

σ n ′ n = (4π) -1 π 0 dθ sin θ 2π 0 dϕ σ n ′ n (θ, ϕ) . (4)
The highly oscillatory character of the σ n ′ n (θ, ϕ) requires a dense grid for the numerical integration. The integration over θ involved the grid points θ i = 90 • , 60 • , 45 • , 30 • , and 0;

the cross sections are symmetric with respect to 90 • . The integration over ϕ used 24 equally spaced points 0 ≤ ϕ j ≤ 345 • with ∆ϕ = 15 • . For θ = 0 only one angle ϕ is necessary. Thus, the total two-dimensional grid included 97 points for non-symmetric molecules. Because of the symmetry with respect to ϕ = 180 • , the analogous grid for symmetric molecules contained only 53 points. The integration scheme used to evaluate Eq.( 4) took into account the special character of the function sin θ σ n ′ n (θ, ϕ) at θ = 0 and 90 • : At θ = 0 the function is zero and the first derivative with respect to θ is finite, whereas for θ = 90 • the function is finite and the derivative is zero.

For comparison with data obtained in I and in the classical calculations we define transition probabilities according to

Q n ′ n = σ n ′ n /σ 0 (5)
where σ 0 is a common cross section. In what follows we will use the value σ 0 = 437.8 Å2 , which is the largest total cross section obtained in the BSA. 9 The main quantity considered

∆E n = n ′ =n Q n ′ n ∆E n ′ n , (6)
where

∆E n ′ n = E n ′ -E n .
An alternative quantity is the downward energy transfer ∆E

(-)
n , for which the summation in Eq.( 6) is limited to n ′ < n; it was the main term considered in I. Both ∆E n and ∆E (-) n

give qualitatively similar descriptions of the energy transfer in ozone.

B. Classical trajectory calculations

An obvious question is how the quantum mechanical energy transfer compares with the vibrational energy transfer obtained from classical trajectories. Details of the classical calculations are described elsewhere. [START_REF] Ivanov | [END_REF]21 For consistency with the quantum mechanical calculations, the ozone molecule is prepared with zero total angular momentum and a specific internal energy E int before the collision. Although O 3 initially does not rotate, during the collision it will be rotationally excited and the rotational energy transfer actually is about one order of magnitude larger than the vibrational energy transfer. 21 In order to compare with the quantum mechanical studies, which consider only vibrational energy exchange, the rotational and the vibrational contributions to the internal energy therefore were separated as described by Ivanov et al. [START_REF] Ivanov | [END_REF] Briefly, the instantaneous tensor of inertia is diagonalized at each point on a trajectory and an instantaneous rotational energy T r is defined using standard expressions. T r is a highly oscillatory function of time. However, averaging over a time interval ∆t and over an ensemble of many trajectories removes the oscillations, which allows to calculate the vibrational energy after the collision according to E int -T r . Because of intramolecular energy flow between the vibrational degrees of freedom, on one hand, and the 'active' rotational degrees of freedom, related to the projection quantum number K a , on the other, the classical vibrational energy is not constant. [START_REF] Ivanov | [END_REF] The rate of this intramolecular energy redistribution is an increasing function of K a as well as the internal energy. However, on the collisional time scale of ∼ 1 ps it can be neglected and a meaningful vibrational energy transfer ∆E cl vib can be extracted. Calculations were performed for several internal energies of the isotopomers 686 and 668.

For each energy between 0.5 × 10 6 and 1.0 × 10 6 trajectories were calculated. In all cases the maximum impact parameter was set to b max = 7 Å which corresponds to a cross section of πb 2 max = 153.9 Å2 . This large impact parameter guarantees that ∆E cl vib is practically zero beyond b max . For comparison with the quantum mechanical ∆E n , however, it is necessary to normalize ∆E cl vib to the same total cross section. Therefore, ∆E cl vib is multiplied with πb 2 max /σ 0 = 0.351, where σ 0 is the cross section used in Eq.( 5).

III. RESULTS

A. Vibrational energy transfer in a single collision

The vibrational energy transfer was calculated for all states up to the dissociation threshold for the isotopomers 666, 686, and 668. ∆E n is negative for the majority of vibrational states in all three isotopomers; at a collision energy of 200 cm -1 , ozone is preferentially de-excited by argon. Despite averaging over many orientation angles, ∆E n in the IOSA is strongly state-specific. The energy transfer fluctuates over two orders of magnitude, and the amplitude of fluctuations is comparable with that in the BSA (Ref. 9). These fluctuations reflect the individual nodal patterns of the vibrational eigenstates, which in ozone are even at high energies mostly assignable. 22 We first discuss the quantum mechanical energy transfer in the 668 isotopomer. Surprisingly, the BSA and the IOSA results are close to each other near the dissociation threshold, where the deviation is smaller than a factor of two. However, the energy transfer in the IOSA is much less energy dependent than in the BSA: The slope of ∆E n vs. E n in the IOSA is substantially smaller than in the BSA over a broad energy range down to about -4000 cm -1 .

This difference in slopes reflects inelastic energy exchange in different groups of vibrational n , both in the IOSA and the BSA, has the same slope as ∆E n . These transitions faithfully describe the exact energy transfer for internal ozone energies up to E n ≤ -2000 cm -1 , and it is this energy range, free from any boundary effects, which we are primarily concerned with. The algorithm to isolate the characteristic transitions m(max) → n ′ (max) has been described in detail in Ref. 9.

The most probable downward transitions in 668 are mapped onto the vibrational spectrum of ozone in Fig. 4. All states with E n ≤ -2000 cm -1 can be assigned vibrational quantum numbers of the symmetric (n s ) and anti-symmetric (n a ) stretch and the bend (n b ). More relevant for ozone with its accidental 1:1 resonance between the symmetric and the antisymmetric modes, however, is the assignment in terms of the polyad quantum number P = n s +n a and the bending quantum number n b (Fig. 4). In the BSA and for energies E n ≥ -4000 cm -1 , transitions occur predominantly between states belonging to different polyads, ∆P = 0 [see Fig. 4(a) and Ref. 9]. The energies of these states are close to each other and are related via a set of resonance conditions which are satisfied by the vibrational frequencies as discussed in I. Below -4000 cm -1 , however, the transitions between adjacent states inside one polyad, ∆P = 0, become the most probable ones. These states are quasi-degenerate, too, because of the near-resonance of the symmetric and the anti-symmetric stretching frequencies. The transition from mainly inter-polyad to mainly intra-polyad energy transfer explains, according to I, the sharp change in the slope of ∆E n vs E n in the BSA around -4000 cm -1 (black thin curve in Fig. 3). Inter-polyad transitions involving simultaneous change of all quantum numbers require higher order anharmonicities in the potential making its matrix elements strongly energy (or quantum number) dependent. In contrast, the intrapolyad transitions, involving merely an incremental change in n a and n s , give rise to the more gently declined branch of ∆E n at lower energies.

The IOSA interaction drastically simplifies the pattern of most probable transitions in 668 [see Fig. 4(b)]. For all energies below -2000 cm -1 , the dominant transitions are exclusively those inside polyads. As a result, the slope of the IOSA energy transfer is consistently smaller than in the BSA for all energies E n ≥ -4000 cm -1 . Below -4000 cm -1 , where the The drastic difference between the state-specific energy transfer dynamics in the two approximations becomes conspicuous in 666 (see Fig. 5). Each polyad in symmetric molecules consists of alternate symmetric and anti-symmetric states with even and odd n a , respectively.

The BSA is a symmetry preserving approximation: The corresponding Ar -O 3 potential is symmetric with respect to interchange of the two end oxygen atoms, and transitions between states of opposite symmetries are forbidden. Thus, all dominant transitions in the BSA are of the inter-polyad type [Fig. 5(a)]. As a consequence, the BSA energy transfer in 666 and likewise 686 is strongly energy dependent throughout the whole ozone spectrum (Fig. 3). Below E n = -4000 cm -1 , this energy dependence gives rise to the pronounced symmetry effect discussed in I. The non-symmetric IOSA interaction reverses the energy transfer dynamics in 666: Symmetry-breaking intra-polyad transitions become dominant at all energies [Fig. 5 As a result, the smoothed IOSA energy transfers in the symmetric (666 and 686) and the non-symmetric (668) isotopomers are nearly equal to each other (Fig. 3).

A quantitative comparison between ∆E n in the IOSA and in the BSA depends on the initial ozone energy. At low E n , the energy transfer in the IOSA is about one order of magnitude larger than in the BSA. However, above E n = -4000 cm -1 the BSA ∆E n increases more rapidly than the IOSA ∆E n and the discrepancy gradually becomes less dramatic. At about -1000 cm -1 the ratio is only two and becomes even smaller for energies closer to the threshold.

Both quantum mechanical approximations show a drastic increase of ∆E n in the energy range about 200 cm -1 below threshold. This is an artifact of the calculations and originates from the restriction of the vibrational basis to bound states only. This was tested by calculations, for one partial wave l, in which the vibrational basis was augmented by the "box states": Extending the vibrational basis to energies above threshold removes the sharp increase of ∆E n near E n ≈ 0. The results for other l values are expected to show the same changes when the basis is augmented and therefore we conclude that the l-averaged energy The IOSA energy transfer agrees over a large range of internal energy very well with ∆E cl vib (Fig. 3). Especially the increase with E n is satisfactorily reproduced by the classical calculations. The comparison between the quantum mechanical and the classical calculations is by no means trivial. First, for the classical results E = 0 corresponds to the classical dissociation threshold while for the quantum mechanical calculations it corresponds to the quantum mechanical threshold; the difference is of the order of 800 cm -1 . Second, ozone is rotationally excited in the classical collisions and the 'active' rotational degrees of freedom are coupled to the vibrational motion. The pure vibrational energy transfer is very small and in the classical calculations it is obtained by subtracting two much larger quantities, E int and T r . This may also be the origin of the relatively large difference between the results for 668 and 686 at very low energies (E n ≈ -6000 cm -1 ). ∆E cl vib seems to exhibit a slight isotope effect: The result for the symmetric molecule is consistently larger than the energy transfer for the non-symmetric one. However, in view of the complexity of the classical calculations it is not justified to further dwell on this difference.

The agreement between the IOSA and the classical calculations is remarkable. We know only one other study in which the energy transfer from highly excited molecules has been calculated by quantum mechanical and classical methods: The collinear collision of CS 2 with He. 23 In that study, which because of lack of rotation is considerably simpler than three-dimensional Ar + O 3 collisions, also perfect agreement between the classical and the averaged quantum mechanical energy transfer has been found.

B. Vibrational energy transfer in multiple collisions

In I we additionally investigated vibrational relaxation of excited ozone via a master equation approach. In a simple finite difference form of the master equation on a discrete grid τ k of the dimensionless time τ = ω 0 t (τ k -τ k-1 = 1) the change of the population of state n is calculated iteratively by

z n (k + 1) = n ′ Q nn ′ z n ′ (k) (7)
("multiple collisions"). The non-diagonal elements Q nn ′ are defined in Eq.( 5) and the diagonal elements are given by

Q nn = 1 -n ′ =n Q n ′ n .
In I, the probabilities were determined

∆E k = n z n (k)∆E n , (8)
where ∆E n is defined in Eq.( 6).

In Fig. 6 we show ∆E k as function of the average ozone energy Ēk for 668 and 686. According to Eq.( 8), ∆E k is an average of the state specific energy transfer in a single collision with Ar. The weighting is provided by the actual populations z n (k), i.e., it is a 'dynamical' weighting. The results for the two isotopomers are basically identical. For comparison we also show the average state specific energy transfer ∆E n obtained by averaging over 20 neighboring initial states. Except for energies above -1500 cm -1 the energy transfer from the multiple collision approach, ∆E k , is smaller than the average first moment ∆E n . The deviations are largest at energies below -4000 cm -1 . The origin of the difference is the different weighting in the averaging process -dynamical weighting in the master equation approach vs. uniform weighting in the single collision procedure -and has been analyzed in I.

C. Simple stabilization/dissociation model

We used the discretized master equation, Eq.( 7), to mimic in a simple way the stabilization and dissociation of highly excited ozone molecules. If Ar collides with ozone molecules in vibrational states close to the dissociation threshold, some fraction of molecules will be de-excited and the remaining part will be excited into the continuum. In the simple model we assume that the molecules excited to continuum states will immediately dissociatei.e., their population is set to zero in the next iteration -and therefore the percentage of stable molecules, P stab (k) = n z n (k), gradually decreases with k; the sum extends over all bound states. In the limit of large k, P stab (k) approaches a constant value and we define this constant as the stabilization probability P stab . It depends on the particular initial state n.

The association of O and O 2 to form highly excited O 3 , i.e., the first step of the Lindemann The most important question concerns the definition of continuum states. We performed two types of calculations. In the first approach we define all bound states above -200 cm -1 as continuum states; in these calculations the cross section based probabilities Q n ′ n (summation over all partial waves, averaging over the full sphere) are used in Eq.( 7). In the second set of calculations the "box states" above the dissociation threshold are considered as continuum, while all bound states are considered to represent stable ozone molecules. In the latter calculations only the transition probabilities for partial wave l = 40, averaged over ϕ for fixed θ = 90 • (in-plane scattering), are considered in Eq.( 7).

The stabilization probabilities for the various initial states n obtained in the two calculations are shown vs. E n in Fig. 7. P stab is very small for states close to the 'threshold' and then gradually rises towards one as the initial energy decreases. Molecules starting with an energy of a few hundred cm -1 below threshold are all stabilized. The energy at which P stab starts to deviate from one depends, of course, on the collision energy. It also depends on the partial wave l, because the inelasticity and therefore the energy transfer depend on l.

The main result of these calculations is that the stabilization probability is independent of the symmetry of the isotopomer: Within the IOSA and the simple stabilization model, symmetric and non-symmetric isotopomers are stabilized with the same probability. Without

showing results we note that the same finding is also obtained when the transition probabilities are calculated in the BSA. This is in accord with the energy transfer per collision, ∆E k , being independent of symmetry in the upper energy regime E n > -1500 cm -1 or so. 9

IV. SUMMARY

The state-resolved vibrational energy transfer ∆E n in Ar -O 3 collisions has been investigated by means of the quantum mechanical infinite order sudden approximation (IOSA).

The study complements our earlier investigation using the computationally much less laborious breathing sphere approximation (BSA). 9 At low ozone energies the IOSA ∆E n is considerably larger than the energy transfer calculated with the BSA; the ratio is about one order of magnitude. Since the BSA energy transfer increases with E n more rapidly than the IOSA ∆E n , the disparity becomes less significant with increasing E n . Around the threshold both approximations predict almost the same vibrational energy transfer. The IOSA results The results for two isotopomers, 686 and 668, have been compared. Within the fluctuations of the averaged state specific ∆E n the results for the symmetric and the non-symmetric isotopomers are essentially the same. This is valid for the IOSA, which does not conserve symmetry, for the entire range of E n and it applies, at least for high energies, also to the BSA, which conserves symmetry. The pronounced symmetry effect found in the BSA at low ozone energies 9 is not relevant for the stabilization of highly excited ozone. Thus, we conclude that the vibrational energy transfer in collisions of highly excited O 3 with Ar does not depend on the symmetry of the isotopomer, at least not when ozone is treated as a simplified C 2v molecule. In cannot be precluded that the results may be different, when all the three main potential wells and the three van der Waals wells are taken into account; but that requires much more demanding calculations.

F

The stabilization process has been modeled with a simplified version of the discretized master equation, in which all states above a certain energy are considered to immediately dissociate. In accord with the independence of the energy transfer on the symmetry of ozone, the stabilization probabilities for the symmetric and the non-symmetric molecules are essentially the same.

3 +

 3 has not been explained up to now. Both, the association/dissociation, O + O 2 ⇋ O * M → O 3 + M, can be affected by symmetry,

10

 10

  is n = 230 with an energy of 213 cm -1 below the threshold. The approach perpendicular to the ozone plane (θ = 0) allows the closest encounter between Ar and O 3 . Approaching one of the end atoms in the plane spanned by O 3 leads to the strongest repulsion (ϕ = 105 •

  is the collision induced vibrational energy transfer (first moment)

  For the presentation, the fluctuations are smoothed by averaging ∆E n over twenty neighboring states. [Throughout the following, ∆E n will always represent the smoothed energy transfer without special notation.] The results for 686 and 668 are depicted in Fig. 3; the data for 666 are similar. In the figure, the IOSA results are compared with the (smoothed) BSA results of Ref. 9 and with the classical trajectory calculations. Despite the averaging the IOSA and BSA energy transfers still exhibit substantial fluctuations. They are slightly more pronounced for the symmetric molecule. The energy zero in Fig. 3 corresponds to the quantum mechanical thresholds in the IOSA and the BSA calculations.

  the BSA, the demonstration of this relation was a focal point of I. The same analysis is directly applicable to the IOSA energy transfer in Fig.3. As in I, we restrict the discussion to the most probable downward transitions m(max) → n ′ (max) which dominate the smoothed downward energy transfer, ∆E (-) n ; the downward energy transfer ∆E(-)

  both approximations occur inside polyads, the slope of the IOSA energy transfer is roughly equal to the slope of the BSA ∆E n .

  (b)] and the slope of the IOSA energy transfer is much smaller than in the BSA. Comparison of the IOSA results for 666 and 668 [cf. Fig. 4(b) and Fig. 5(b)] shows that the most probable downward energy flow is almost indistinguishable in both molecules.

  near threshold -like the classical vibrational energy transfer.

  temperature by averaging over the collision energy E c according to the Maxwell distribution. Because the IOSA calculations are considerably more time consuming, the Q nn ′ in the present work are calculated for a fixed collision energy E c = 200 cm -1 . The initial distribution z n (0) includes a few of the uppermost states. Repetition of Eq.(7) results in the relaxation of the average energy Ēk = n z n (k)E n . The energy relaxation rate ∆E k = Ēk -Ēk+1 is given by

  the vibrational energy transfer calculated by classical trajectories over the entire range of internal energies. Within this range, the vibrational energy transfer varies by more than an order of magnitude. To our knowledge, this is the first comparison of quantum mechanical and classical vibrational energy transfer in full-dimensional collisions of a highly excited polyatomic molecule with an atom.

FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :FIG. 5 :FIG. 6 :

 123456 FIG.1: Elastic potential matrix elements V nn (ρ) for state n = 230 of isotopomer 686 and several angles (θ, ϕ). The innermost curve is for θ = 0 • (perpendicular approach) while the other curves are for θ = 90 • (in-plane collisions): ϕ = 30 • (inner curve), 180 • (middle curve), and 105 • (outer curve).

17 FIG. 7 :

 177 FIG. 7: Stabilization probability P stab vs. ozone energy. (a) Calculations with all bound states above -200 cm -1 considered as continuum states; the vertical dashed line indicates the fictitious threshold. (b) Calculations including "box states" in order to simulate the continuum. Further details are given in the text. Page 17 of 24

Fig. 1 Page

 1 Fig.1

C. G. Schatz and G. Lendvay, J. Chem. Phys. 106, 3548 (1997).

Acknowledgments

Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. The authors are grateful to S. Yu. Grebenshchikov for invaluable discussions and contributions. Especially the analysis of the energy transfer in terms of the polyad structure of the ozone spectrum in Sect. III.A and the related Figs. 4 and 5 are his work. To the regret of