Ken Op De Beeck

Guy Van Camp
email: guy.vancamp@ua.ac.

Sofie Thys

Nathalie Cools

Isabelle Callebaut

Karen Vrijens

Luc Van Nassauw

Viggo Fi Van Tendeloo

Jean Pierre Timmermans

Lut Van Laer

Viggo Fi

Van Tendeloo

The DFNA5 gene, responsible for hearing loss and involved in cancer, encodes a novel apoptosis-inducing protein

Keywords: tumor suppressor, hearing loss, apoptosis, dfna5, cancer, hca, GSEA -2 -

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

DFNA5 first was discovered in a Dutch family with autosomal dominant hearing loss (HL) [START_REF] Van Laer | Nonsyndromic hearing impairment is associated with a mutation in DFNA5[END_REF] .

Not much was known concerning its cellular function and how its function was related to HL.

Recently, a novel mutation in DFNA5 has been identified in a Korean family, totaling 5 families with DFNA5 hearing loss [START_REF] Park | Evidence for a founder mutation causing DFNA5 hearing loss in East Asians[END_REF] . These families all have different genomic DFNA5 mutations, but in each case the DFNA5 mRNA transcript skips exon 8, resulting in a frameshift and a premature truncation of the protein [START_REF] Van Laer | Nonsyndromic hearing impairment is associated with a mutation in DFNA5[END_REF][START_REF] Park | Evidence for a founder mutation causing DFNA5 hearing loss in East Asians[END_REF][START_REF] Cheng | A novel DFNA5 mutation, IVS8+4 A>G, in the splice donor site of intron 8 causes late-onset non-syndromic hearing loss in a Chinese family[END_REF][START_REF] Bischoff | A second mutation identified in the DFNA5 gene in a Dutch family. A clinical and genetic evaluation[END_REF][START_REF] Yu | A 3-nucleotide deletion in the polypyrimidine tract of intron 7 of the DFNA5 gene causes nonsyndromic hearing impairment in a Chinese family[END_REF] . These findings have led to the hypothesis that DFNA5 associated HL is attributable to a highly specific gain-of-function mutation, in which skipping of one exon causes disease while mutations in other parts of this gene may not result in HL at all. Further experimental evidence for this hypothesis was provided by the finding that transfection of mutant DFNA5 causes cell death in both yeast [START_REF] Gregan | A yeast model for the study of human DFNA5, a gene mutated in nonsyndromic hearing impairment[END_REF] and mammalian [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] cells and by the discovery of a new DFNA5 mutation [START_REF] Van Laer | A novel DFNA5 mutation does not cause hearing loss in an Iranian family[END_REF] . The latter mutation truncated the protein in the fifth exon, but did not segregate with HL and was present in family members with normal hearing. The hypothesis was further corroborated by a mouse that lacked the Dfna5 protein. This knockout (KO) mouse did not display any HL and, as a consequence, was not a suitable animal model to study DFNA5 associated HL [START_REF] Van Laer | Mice lacking Dfna5 show a diverging number of cochlear fourth row outer hair cells[END_REF] .

To date, little information is available on the physiological function of DFNA5. However, since its identification, the small number of papers published on DFNA5 almost all point to a possible involvement in cancer biology [START_REF] Thompson | Characterization of a gene that is inversely correlated with estrogen receptor expression (ICERE-1) in breast carcinomas[END_REF][START_REF] Lage | DFNA5 (ICERE-1) contributes to acquired etoposide resistance in melanoma cells[END_REF][START_REF] Masuda | The potential role of DFNA5, a hearing impairment gene, in p53-mediated cellular response to DNA damage[END_REF][START_REF] Akino | Identification of DFNA5 as a target of epigenetic inactivation in gastric cancer[END_REF][START_REF] Kim | Aberrant promoter methylation and tumor suppressive activity of the DFNA5 gene in colorectal carcinoma[END_REF][START_REF] Kim | Methylation of the DFNA5 increases risk of lymph node metastasis in human breast cancer[END_REF][START_REF] Fujikane | Genomic screening for genes upregulated by demethylation revealed novel targets of epigenetic silencing in breast cancer[END_REF] . Especially during the last three years, DFNA5 emerged from several genomic screens identifying new tumor suppressor genes. Further analysis showed that the DFNA5 gene is epigenetically inactivated in several types of cancer, including gastric, colorectal and breast cancer [START_REF] Akino | Identification of DFNA5 as a target of epigenetic inactivation in gastric cancer[END_REF][START_REF] Kim | Aberrant promoter methylation and tumor suppressive activity of the DFNA5 gene in colorectal carcinoma[END_REF][START_REF] Kim | Methylation of the DFNA5 increases risk of lymph node metastasis in human breast cancer[END_REF][START_REF] Fujikane | Genomic screening for genes upregulated by demethylation revealed novel targets of epigenetic silencing in breast cancer[END_REF] . Epigenetic silencing through methylation was found in 52 to 65% of primary tumors. In addition, in vitro studies showed that cell invasion, colony numbers, colony size and cell growth increased in cell lines after DFNA5 knockdown. Forced expression of DFNA5 in colorectal carcinoma cell lines, on the other hand, decreased cell growth and colony forming ability. In breast carcinoma, the methylation status of DFNA5 was correlated with lymph node metastasis [START_REF] Kim | Methylation of the DFNA5 increases risk of lymph node metastasis in human breast cancer[END_REF] . These recent data also explain earlier findings that were not well understood at the time, such as the increased resistance to the chemotherapeutic reagent etoposide in melanoma cells with decreased transcription of DFNA5 [START_REF] Lage | DFNA5 (ICERE-1) contributes to acquired etoposide resistance in melanoma cells[END_REF] , or the finding that DFNA5 is induced by p53 through a p53-binding site located in intron 1 [START_REF] Masuda | The potential role of DFNA5, a hearing impairment gene, in p53-mediated cellular response to DNA damage[END_REF] . of human acute lymphoblastic leukemia cell lines with the glucocorticoid dexamethasone. This increase was only found in cells that underwent apoptosis in response to glucocorticoid treatment [START_REF] Webb | In CEM cells the autosomal deafness gene dfna5 is regulated by glucocorticoids and forskolin[END_REF] . The conclusion of these data is that DFNA5 is a tumor suppressor gene with an important role in several frequent forms of cancer.

Here we report that DFNA5 contains two domains separated by a hinge region and that the first domain induces apoptosis when transfected into cell lines. The second domain may shield the apoptotic-inducing sequences residing in the first domain. Furthermore, using gene expression microarray experiments on Dfna5 KO mice, we provide evidence that the apoptosis inducing properties of DFNA5 also occur in a physiological setting and that DFNA5 is involved in cell survival pathways.

Materials and methods

Plasmid construction

Using previously described human full-length DFNA5 cDNA clones as a template [START_REF] Gregan | A yeast model for the study of human DFNA5, a gene mutated in nonsyndromic hearing impairment[END_REF] , we amplified full-length human wild type (WT) and mutant DFNA5 sequences in addition to several specific parts of the DFNA5 cDNA sequence using the iProof High Fidelity PCR kit (Bio-Rad Laboratories, Hercules, CA, USA) according to the manufacturer's instructions. A kozak consensus sequence (GCCACCATG) was introduced in the specific parts that lacked the regular DFNA5 startcodon, either via PCR or via site-directed mutagenesis (Stratagene, La Jolla, CA, USA). PCR primers are listed in supplementary table I. All PCR fragments were ligated in pEGFP-N1. To clone the part of Pejvakin (PJVK) that showed high homology to DFNA5, human testis Quick-clone cDNA (BD Biosciences, San Jose, CA, USA) was used as a template. All cloned inserts were bidirectionally sequenced on an ABI 3730xl genetic analyzer (Applied Biosystems, Foster City, CA, USA).

Transfection experiments

HEK293T, HELA and MCF7 cells were subcultured in 60 mm dishes at a density of 2 x 10 5 cells in Dulbecco's modified Eagle's medium (DME) containing 4500 mg/l glucose supplemented with 10% (volume/volume) fetal calf serum, 100 U/ml penicillin, 100 µg/ml streptomycin and 2 mM L-glutamine (all products from Invitrogen, San Diego, CA, USA).

The cells were incubated overnight at 37°C in a 5% CO 2 humidified environment.

Transfections were performed with a lipofectamine-plasmid emulsion as described previously [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . 21 hours post transfection (16 hours for apoptosis assays; only for HEK293T cells), cells were harvested using TrypLE™ Express (Invitrogen) and diluted in 2 ml DME. Following centrifugation, the resulting cell pellet was resuspended in 500 µl D-PBS (Dulbecco's Phosphate-buffered Saline) (Invitrogen) and was subsequently used for flow cytometric evaluation.

Cell viability measurements

Before measurements, 1 µl propidium iodide (PI) (1mg/ml) (Invitrogen) was added to the cell suspensions to enable discrimination between viable and dead cells. Cells were then evaluated on a FACScan flow cytometer (BD Biosciences). The gating strategy and cell counting was as described previously [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . All measurements were independently replicated at least three times, unless stated otherwise. Cell viability was determined as the ratio of cells showing no PI fluorescence to the total cell population.

Confocal microscopy

Cells for confocal microscopy were prepared as described previously [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . Subcellular localization of selected constructs was performed using a Zeiss CLSM 510 confocal laser scanning microscope (Zeiss, Göttingen, Germany) equipped with an argon laser (excitation at 488 nm) and two helium/neon lasers (excitation at 543nm and 630nm, respectively). To test whether the mutant DFNA5 exon 9 -exon 10 construct was localised in the endoplasmic reticulum, double transfection with this construct and with pDsRed2-ER (Clontech) was performed in HEK293T cells. All images were taken at room temperature (RT) using a Zeiss C-Appochromat 63x/1.2 W Korr lens with Zeiss LSM 510 acquisition software.

-6 -

Western Blotting

HEK293T cells were subcultured at a density of 4 x 10 5 cells per well and lipofected as described previously [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . After 21 hours of incubation, cells were lysed on ice using RIPA buffer (150 mM NaCl, 50 mM Tris-HCL, 1% NP40, 0.5% Na-deoxycholate, 0.1% SDS), containing Complete Mini ® protease inhibitors (Roche, Basel, Switzerland). The cell lysate was diluted in sample buffer (60 mM Tris-HCl, 50% glycerol, 2% SDS, 14.4 mM 2mercaptoethanol, 0.1% Bromophenol Blue), heated for 5 minutes at 95°C, loaded on a 12% Tris-Glycine gel (Anamed, Gross-Bieberau, Germany) and electrophoretically separated.

Next, the membrane was incubated with a primary rabbit anti-GFP antibody (1/2500) (Sigma, Missouri, USA) and a secondary goat anti rabbit-horse radish peroxidase-conjugated antibody (1/10000) (Sigma) afterwards. Subsequently, EGFP-fusion proteins were detected using Pierce ECL western blotting substrate (Thermo Scientific, Rockford, IL, USA).

Annexin V staining

HEK293T cells were lipofected as described above. The standard harvesting time for the annexin V experiment was 16 hours post transfection. For the kinetic experiment harvesting times were: 3, 6, 9, 12, 16 and 24 hours post transfection. After harvesting, the cells were stained for annexin V using annexin V-biotin (Roche) according to the manufacturer's protocol. Before measurements, cells were stained with 1.3 µg/ml PI. Using a CyFlow ML cytometer (Partec, Görlitz, Germany), viable, early apoptotic, late apoptotic and necrotic cells were analyzed. Transfected annexin V-stained cells were also visualized using confocal microscopy.

TUNEL assay

HEK293T cells were transfected as described above, harvested after 16 hours and a TUNEL assay ("in situ cell death detection kit", tetramethylrhodamine (TMR) red (Roche)) was performed according to the manufacturer's instructions. Cells were then resuspended in D--7 -PBS (Invitrogen), cytospinned and enclosed in Citifluor (Ted Pella, Redding, CA, USA).

Cytospins were studied using confocal microscopy. Random images were taken at RT with Zeiss LSM 510 acquisition software on a Zeiss CLSM510 Meta microscope using a Zeiss C-Appochromat 40x/1.2 W Korr lens. For each experiment, the total number of TUNEL positive cells and the total number of cells were counted from 10 random images, using the AnalySIS software.

Microarray expression analysis of Dfna5 KO versus WT mice and subsequent real time PCR verification

Inner ear samples were removed from 9 WT (+/+) postnatal day 0 (P0) and 9 Dfna5 KO (-/-) (P0) C57Bl/6J mice and mechanically homogenized, after which total RNA was extracted using the RNeasy kit (Qiagen, Hilden, Germany) according to the manufacturer's instructions.

RNA quality was verified using the Experion system (Biorad). Subsequently, 9 RNA samples from WT mice were pooled in 3 WT RNA pools and 9 RNA samples from KO mice were pooled in 3 KO RNA pools. RNA pools were labeled with either Cy3 or Cy5 dyes, and hybridized on a Whole Mouse Genome Array (41K array) (Agilent, Santa Clara, CA, USA).

Each experiment was followed by a colourflip, resulting in a total of 6 microarray experiments. The resulting intensity data was analyzed using the R-package "limma" (v3.2.1) and the Gene Set Enrichment Analysis (GSEA) program (v2.0, Broad Institute) [START_REF] Subramanian | Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles[END_REF] . Gene sets for GSEA were downloaded from the Gene Ontology (GO) website (http://www.geneontology.org/). These microarray results were validated with real time PCR using the RNA pools from the microarray experiment and a one step Power SYBR green PCR master mix (Applied Biosystems) according to the manufacturer's instructions on a Lightcycler 480 instrument (Roche). The resulting gene expression data were analyzed using Qbase plus (Biogazelle, Gent, Belgium) software.

Statistical Analysis

All statistical analyses were performed using one-way ANOVA. P values below 0.05 were considered statistically significant. Statistics were calculated using SPPS 15.0 (SPSS inc, Chicago, IL, USA)

Results

Gross mapping of the toxic region

All DFNA5 mutations that cause hearing loss lead to exon 8 skipping. Therefore, we tested whether the toxicity of the mutant protein was due to the aberrant tail of 41 amino acids (AA) that is present in the mutant protein, as the first seven exons are identical in WT and mutant DFNA5 [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . We cloned the first part of DFNA5 (exon 2 -exon 7) and generated three constructs based on the second part of DFNA5 (exon 8 -exon 10 and exon 9 -exon 10 in the WT as well as in the mutant reading frame (figure 1A)) in a pEGFP-N1 vector. The expression of the fusion proteins was confirmed by western blotting (figure 1B). Figure 1C The subcellular localization of these EGFP-tagged proteins was determined (figure 1D).

Transfection of the WT DFNA5 exon 2 -exon 7 construct resulted in a fusion protein that was expressed mainly at the plasma membrane. In addition, it was observed in the cytoplasm in a granulated manner. The majority of cells transfected with this construct showed an unhealthy appearance (i.e., cell blebbing). Overall, the expression pattern was very similar to that of mutant DFNA5 [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . WT DFNA5 exon 8 -exon 10 and WT DFNA5 exon 9 -exon 10 exhibited a highly similar expression pattern. Both fusion proteins were distributed along the cytoplasm and cluster near the nucleus. Finally, cells transfected with mutant DFNA5 exon 9 -exon 10 showed EGFP expression primarily in the endoplasmic reticulum (ER). This was confirmed by cotransfection experiments with pDsRed2-ER, in which the EGFP signal of mutant DFNA5 exon 9 -exon 10 colocalized with the signal of DsRed (supplementary figure 1).

Hydrophobic Cluster Analysis (HCA)

Simultaneously with the experiments described above, the sequence of DFNA5 was analyzed through HCA and compared to those of the DFNA5-gasdermin family, to which DFNA5 belongs (figure 2). This technique of protein sequence analysis allows the simultaneous comparison of primary and secondary structures [START_REF] Callebaut | Deciphering protein sequence information through hydrophobic cluster analysis (HCA): current status and perspectives[END_REF][START_REF] Eudes | A generalized analysis of hydrophobic and loop clusters within globular protein sequences[END_REF][START_REF] Gaboriaud | Hydrophobic cluster analysis: an efficient new way to compare and analyse amino acid sequences[END_REF] .

From the comparison of DFNA5-gasdermin family members, DFNA5 can be divided in three distinct domains: a globular domain A (exon 2 -exon 6), a hinge region (exon 6 -exon 7) which is variable in length and sequence among the different DFNA5-gasdermin members and a globular domain B (exon 7 -exon 10). Domain A, which is common to all members of the DFNA5-gasdermin family, had an α/β fold, whereas domain B, which is present in GSDM1 and MLZE but not in PJVK (in this case, it is replaced by a zinc-finger domain), had an all α fold (mainly α-helical structures). The lengths and shapes of the hydrophobic clusters within domain B suggested the presence of long helical structures, which might form coiledcoils.

Based on the division proposed by HCA, we constructed two plasmids, the first one containing HCA part A and the second HCA part B. When transfected in HEK293T cells, domain A but not domain B was able to induce a toxic effect when compared to WT DFNA5 (see supplementary table II). This finding enabled us to exclude exon 7 as part of the toxic region since this exon was not present in domain A of DFNA5. Domain A of DFNA5 is highly similar to the first part of PJVK. To investigate whether the mechanism of toxicity was a conserved feature among the two members of the DFNA5 family, we cloned the first part of PJVK (as indicated by HCA) in the pEGFP-N1 vector.

After overexpression of the first part of PJVK in HEK293T cells, we measured no viability drop (see supplementary table II).

Further delineation of the toxic region

The toxic region of DFNA5 was further delineated through the generation of different constructs. In a first series of constructs, DFNA5 was truncated exon per exon starting from exon 7 towards the N-terminal region (supplementary figure 2). In another series of constructs DFNA5 was truncated exon per exon from the N-terminal region towards exon 7 (supplementary figure 3). The conclusion of these series of experiments was that an essential part of the toxic DFNA5 region was located somewhere in exon 6 and that the presence of exon 2 was essential for the toxic effect.

The toxic motif did not consist of a single exon as was demonstrated by the transfection of DFNA5 exons 2 to 7 separately (see supplementary table II).In addition, transfection of constructs similar to WT DFNA5 HCA domain A, but lacking specific exons (either lacking exon 3, exon 4 or exon 5) did not lead to a detectable viability loss (see supplementary table II), indicating that the complete region between exon 2 and 6 is required for correct folding of the toxic domain.

Cell Death evaluation.

The fate of cells transfected with mutant DFNA5 was investigated using two independent assays: annexin V staining and TUNEL labeling.

-11 -

Annexin V staining

To distinguish between apoptotic and necrotic cells, we performed a double labeling, using annexin V and PI. Early apoptotic cells were positive for annexin V, but not for PI. Late apoptotic cells were positive for both annexin V and PI (due to secondary necrosis), while necrotic cells emitted fluorescent signals only from PI.

Cells were transfected with an empty EGFP vector and with WT and mutant DFNA5-EGFP constructs and were visualized using confocal laser microscopy (figure 3A). The vast majority of cells transfected with WT DFNA5, showed no signs of PI or annexin V staining, while cells transfected with mutant DFNA5 exhibited clear annexin V-Cy5 staining at their plasma membranes. Since no PI staining could be detected, this finding clearly indicated that cell death caused by mutant DFNA5 was due to an apoptotic process.

To obtain quantitative results, we stained transfected HEK293T cell suspensions with annexin V-Cy5. Prior to flow cytometric analysis we also added PI to the cell suspensions. This allowed to discriminate between the viable, early apoptotic, late apoptotic and necrotic fractions. A typical flow cytometric result is shown in supplementary figure 4. To obtain the total apoptotic fraction in each cell suspension, we added the early apoptotic to the late apoptotic fraction. The bar charts in figure 3B summarize the average total apoptotic fraction (3 independent measurements) for each transfected cell suspension. Cells transfected with mutant DFNA5 had a higher total apoptotic fraction (31.87%±7.82) than cells transfected with WT DFNA5 (18.32%±5.51). Although the difference between the two groups was not statistically significant (p=0.07), the trend was clearly visible.

TUNEL Assay

We transfected HEK293T cells with WT DFNA5 and mutant DFNA5-EGFP constructs and an empty EGFP vector as negative control. Further controls included an untransfected and a mock transfected cell population. After staining, we counted the average TUNEL-positive cells in 10 randomly captured images and made a ratio of these positive cells to the total number of cells present in these images (figure 4). A statistically significant difference (p=0.02) was observed between mutant DFNA5 transfected cells and cells transfected with WT DFNA5. The combined results of TUNEL and annexin V staining clearly indicate that toxicity induced by mutant DFNA5 was due to apoptosis.

Microarray experiments using WT and Dfna5 KO mice

To evaluate whether the apoptosis inducing features of DFNA5 could also be demonstrated in a physiological setting, we performed a microarray gene expression analysis on WT and Dfna5 KO mice using the Agilent Whole Mouse Genome Array (41k). Differential gene expression was estimated and 92 genes were statistically significant upregulated (adjusted p value less than 0.05) in KO mice, while 88 genes were statistically significant downregulated in KO mice when compared to WT mice. From this list, four genes were selected and their expression patterns were evaluated using real time PCR and compared to those obtained from the microarray study. The differential expression of these genes was very similar between the two techniques (supplementary figure 5 and supplementary table III), indicating that the microarray results were reliable. Next, a gene set enrichment analysis (GSEA) was performed to check whether specific biological pathways were up-or downregulated in these mice to extract biologically meaningful conclusions.

Discussion

In the current study, we showed that induction of apoptosis is an intrinsic feature of the physiological function of DFNA5. We previously assumed that transfection of mutant DFNA5 causes necrotic cell death [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . Since increasing evidence of tumor suppressor gene characteristics of DFNA5 has been provided [START_REF] Lage | DFNA5 (ICERE-1) contributes to acquired etoposide resistance in melanoma cells[END_REF][START_REF] Masuda | The potential role of DFNA5, a hearing impairment gene, in p53-mediated cellular response to DNA damage[END_REF][START_REF] Kim | Aberrant promoter methylation and tumor suppressive activity of the DFNA5 gene in colorectal carcinoma[END_REF] , we re-evaluated the nature of mutant DFNA5-induced cell death. In this report, we used annexin V as well as TUNEL staining.

Both techniques showed that transfection of mutant DFNA5 resulted in cell death due to apoptosis. A likely explanation for the misinterpretation 7 may be that the time-lapse between transfection and assaying for cell death was too long in our previous report. We previously waited 24 hours before harvesting the cells [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] , versus 16 hours in the current study. We further investigated this concept with an experiment to establish the kinetics of mutant DFNA5induced cell death. In a kinetic experiment, we harvested cells transfected with WT and mutant DFNA5 after 3, 6, 9, 12, 16 and 24 hours (results not shown). Apoptotic events, identified as a positive signal for annexin V staining, became evident as soon as 6 hours after transfection with mutant DFNA5, and remained evident up to the time point of 16 hours, our standard harvesting time. Cells that were harvested after 24 hours had already shifted from late apoptotic cell death to secondary necrotic cell death.

In this study, we localized two essential sections of the apoptosis-inducing domain. Our results demonstrate that the apoptosis-inducing domain is not merely a linear domain located in a single or a few exons, but a three-dimensional structure consisting of different noncontiguous parts of the protein, and that the complete region between exon 2 and exon 6 may be critical for the correct formation of the apoptosis-inducing motif. This implicates that proper folding of the globular domain formed by the region including exon 2 to exon 6 is essential.

The apoptosis-inducing mechanism of DFNA5 is not limited to transfections of HEK293T cells. Previously, we showed that transfection of mutant DFNA5 results in a viability loss in COS cells [START_REF] Van Laer | DFNA5: hearing impairment exon instead of hearing impairment gene?[END_REF] . Here, we extended our analysis and transfected additional cell lines (MCF7 and HELA) with mutant and WT DFNA5 (see supplementary figure 6). It appeared that transfection efficiencies in these cell lines were much lower than those obtained in HEK293T.

Hence, transfection of mutant DFNA5 did not impact viability of the overall cell population.

However, when only GFP positive subpopulations were taken into account, it was clear that mutant DFNA5 causes a significant drop in viability which is not seen after transfection of WT DFNA5. In conclusion, the apoptosis inducing mechanism of DFNA5 is not limited to transfections in HEK293T, but is present in all investigated cell lines so far.

Gene expression microarray experiments on WT and Dfna5 KO mice, followed by pathway analysis revealed that apoptosis inducing pathways are downregulated in Dfna5 KO mice. As such, we provide the first experimental evidence that the apoptotic features of DFNA5 are not only limited to overexpression experiments, but are also present in a physiological setting. Additionally, we have observed an upregulation of the DNA repair mechanisms in Dfna5 KO mice. The latter could be explained by the fact that when apoptotic pathways are downregulated, DNA repair mechanisms need to be upregulated to maintain viable conditions.

Interestingly, genes involved in cartilage maintenance were also upregulated in Dfna5 KO mice, linking our findings to those obtained by Busch-Nentwich et al. [START_REF] Busch-Nentwich | The deafness gene dfna5 is crucial for ugdh expression and HA production in the developing ear in zebrafish[END_REF] . They found that zebrafish injected with Dfna5 morpholino's displayed an absent expression of Ugdh, (UDPglucose dehydrogenase) leading to reduced production of hyaluronic acid resulting in impaired facial cartilage differentiation. As Dfna5 KO mice do not display this phenotype, it is possible that the gene sets involved in cartilage development provide compensatory mechanisms resulting in the lack of phenotype in these mice.

-15 -DFNA5 and PJVK clearly form a separate group within the DFNA5-gasdermin family [START_REF] Delmaghani | Mutations in the gene encoding pejvakin, a newly identified protein of the afferent auditory pathway, cause DFNB59 auditory neuropathy[END_REF] , while the other group includes GSDM1, MLZE and homologues. Although the two families evolved from a common ancestor, phylogenetic analysis has indicated that the two clades are distinct [START_REF] Tamura | Members of a novel gene family, Gsdm, are expressed exclusively in the epithelium of the skin and gastrointestinal tract in a highly tissue-specific manner[END_REF] , leaving DFNA5 with only a single closely related protein: PJVK. The first part of PJVK shows high similarity with DFNA5, but does not induce apoptosis. This is a first indication that the apoptosis-inducing mechanism may be specific for DFNA5 and that it may not be a general feature of the DFNA5 protein family.

Still, some questions remain to be clarified. Given the finding that the apoptosis-inducing domain is present in both WT and mutant DFNA5, the question arises as to why transfections with WT DFNA5 did not result in cell death. A plausible explanation could be that the last part of the protein may quench or shield the part in which the apoptosis-inducing domain resides, so that apoptosis is not expressed in normal cellular conditions. Because the mutation changes and shortens the last part of the protein (starting from exon 8), the shielding function of this part of the protein may be neutralized and, as a consequence, apoptosis may be unleashed. This remains to be further investigated, for example by the resolution of the crystal three-dimensional structure of DFNA5.

Methylation of the 5'-flanking region of DFNA5, as frequently observed in gastric, colorectal and breast carcinoma [START_REF] Akino | Identification of DFNA5 as a target of epigenetic inactivation in gastric cancer[END_REF][START_REF] Kim | Aberrant promoter methylation and tumor suppressive activity of the DFNA5 gene in colorectal carcinoma[END_REF][START_REF] Kim | Methylation of the DFNA5 increases risk of lymph node metastasis in human breast cancer[END_REF][START_REF] Fujikane | Genomic screening for genes upregulated by demethylation revealed novel targets of epigenetic silencing in breast cancer[END_REF] may be a way to prevent the apoptotic actions of DFNA5. We believe that our description of the apoptosis-inducing mechanism of DFNA5 contributes to the growing evidence that DFNA5 may be a tumor suppressor gene. As apoptosis induction seems to be an intrinsic feature of DFNA5's function, it is likely that the WT DFNA5 protein can be activated to induce apoptosis in the cell under certain conditions. The nature of this regulating mechanism however, remains to be elucidated. DFNA5-associated hearing loss is nonsyndromic and has so far not been associated with other symptoms. Still, DFNA5 is present in every tissue investigated so far, albeit at low levels of expression. In our opinion, it is unlikely that massive apoptotic cell death takes place in cochleae of DFNA5 mutation carriers, or in any of the other tissues where DFNA5 is expressed. A more plausible scenario for the in vivo situation is that expression of mutant DFNA5 is a pro-apoptotic factor in the cell and that especially terminally differentiated cells, such as cochlear hair cells, are vulnerable to mutant DFNA5-induced programmed cell death. This cochlear hair cell loss may then be manifested as hearing loss. The fact that DFNA5associated hearing loss is progressive in nature, seems compatible with this hypothesis. The role of apoptosis in cochlear cells resulting in auditory pathology has been described previously. For example, it has been demonstrated that apoptosis occurs in outer hair cells in mice displaying age related hearing loss [START_REF] Sha | Activation of cell death pathways in the inner ear of the aging CBA/J mouse[END_REF] . Recently, a report was published implicating apoptosis in monogenic hearing loss. Overexpression of TJP2 leads to altered expression of apoptosis related genes, ultimately causing hearing loss [START_REF] Walsh | Genomic duplication and overexpression of TJP2/ZO-2 leads to altered expression of apoptosis genes in progressive[END_REF] . We believe that our study provides a new line of evidence supporting an important role of apoptosis in hearing loss.

In conclusion, we have demonstrated that DFNA5 is an apoptosis-inducing protein. In addition, it becomes increasingly clear that DFNA5 not only functions in the auditory pathway, but is probably also implicated in a more fundamental role involving cell survival and apoptosis. nonsyndromic hearing loss DFNA51. American Journal of Human Genetics; 87: 101-109.

27.

Woodcock S, Mornon JP, Henrissat B: Detection of secondary structure elements in proteins by hydrophobic cluster analysis. Protein Eng 1992; 5: 629-635.

Titles and legends to figures

  clearly demonstrates that transfection of the first part of DFNA5 in HEK293T cells causes a significant drop in cell viability when compared to transfection of WT DFNA5 (p<0.001). This was an interesting finding because this part of the protein is shared by WT and mutant protein. None of the constructs of the second part of DFNA5 induced cytotoxicity. The C-terminal part of DFNA5 does not have an inhibitory effect on the cytotoxicity displayed by the N-terminal part of DFNA5, as the viability after double transfections with the WT DFNA5 exon 2 -exon 7 and the WT DFNA5 exon 9 -exon 10 construct or with the WT DFNA5 exon 2 -exon 6 and the WT DFNA5 exon 8 -exon 10 construct was comparable to that observed when transfecting the WT DFNA5 exon 2 -exon 7 construct alone (results not shown).

Figure 1 :

 1 Figure 1: Panel A Graphical representation of the insert content in relation to the construct's name. An exon is indicated by a rectangle containing the relevant exon number. The alternative reading frame used in the aberrant tail of mutant DFNA5 is indicated with an asterisk. Panel B Western blot analysis of selected DFNA5 constructs demonstrating that all constructs are expressed when transfected in HEK293T cells (using a pEGFP-N1 vector). Arrows point to the relevant position of the fusion proteins. Some degradation products are visible in lanes 2, 3 and 5. Panel C Cell viability measurements 21 hours post transfection. Graphical bars represent the percentage of viable cells. An asterisk denotes statistically significant differences (p<0.05) between transfections with the construct of interest and the WT construct. Control measurements are depicted in light gray colored bars, while constructs specific for this experiment are indicated with hatched bars. This series of transfections demonstrates that the toxic part of DFNA5 is located in the first part of DFNA5. Interestingly, this indicates that the motif that is responsible for toxicity is shared by WT and mutant DFNA5. Panel D

Figure 2 :

 2 Figure 2: HCA analysis of DFNA5, GSDM1, MLZE and PJVK. The sequences are shown on a duplicated α-helical net, in which hydrophobic residues (VILFMYW) are contoured, forming clusters, which mainly correspond to the internal faces of regular secondary structures 27 . The way to read the sequences (1D) and secondary structures (2D) as well as special symbols are indicated in the inset. Conserved hydrophobic residues are shaded in grey, the vertical lines indicating the correspondences between the different sequences. Noticeable identities/similarities outside the hydrophobic clusters are encircled in grey. The predicted globular domains (A and B) are depicted with arrows. A hinge of variable length (grey box) separates domain A from domain B. Predicted secondary structures of these globular domains are indicated at the top. Domain B does not exist in PJVK, in which it is replaced by a shorter domain (contoured in grey), likely containing a zinc finger. A full color version of this figure can be found in the html version of this paper.

Figure 3 :

 3 Figure 3: Panel A

  The analysis revealed 73 upregulated gene sets and 225 downregulated gene sets in KO mice (pathways with a False Discovery Rate (FDR) < 25%; see supplementary table IV and V for a list of these pathways). Particularly, gene sets involved in cartilage maintenance (indicated in red in supplementary table IV) and DNA repair (indicated in blue in supplementary table IV) are upregulated in Dfna5 KO mice, while gene sets involved in energy metabolism (indicated in orange in supplementary table V) and apoptosis (indicated in purple in supplementary table V) are downregulated in Dfna5 KO mice.

Acknowledgements

This work was supported by grants of the European Community (Eurohear) and the 'Fonds voor Wetenschappelijk Onderzoek Vlaanderen' (FWO grant G.0245.10N). K.O.D.B. holds a predoctoral research position with the 'Instituut voor de Aanmoediging van Innovatie door Wetenschap en Technologie in Vlaanderen (IWT)'.

Conflict of interest

The authors declare no conflict of interest.

Supplementary information is available at the European


L Meissner

Multi-reference many-body perturbation theory and coupled cluster developments

Keywords: many-electron systems, electron correlation, quasi-degenerate states, multi-refernce perturabation theory, multi-reference coupled cluster method

Multi-reference methods for the description of quasi-degenerate ground states and excited states of many-electron systems have been one of the main objectives of scientific activity at the Quantum Theory Project (QTP) University of Florida in Gainesville.

The pioneering works by Dr. Per-Olov Löwdin on the partitioning technique, important contributions to the development of multi-reference methods made by Dr. Monkhorst and Dr. Bartlett must be mentioned here. Dr. Bartlett, his group and its visitors have been especially active in this field. In this paper we would like to describe some of multi-reference developments, problems that have been faced and solved as well as those which still require solving. The focus is on multi-reference many-body perturbation theory and coupled-cluster approaches. We present recent advances and future perspectives in the development of these methods. A special attention is paid to two multi-reference approaches that seem very promising and have been either initiated or developed at QTP.

these determinants should be treated on the same footing in the expansion. Thus they should be all used as references for selecting contributions to the wave function. According to this strategy the single-reference expansions with singles and doubles (SD) should be extended to include also single and double excitations with respect to other dominant determinants since this is more effective then to increase the excitation rank in the SR approaches.

The idea of replacing single reference with a set of reference determinants leads to multireference (MR) formulations of methods designated for description of many-body systems.

There was a hope that with more general formalisms it would be possible to overcome limitations of the single-reference approaches and obtain methods of universal applicability.

Unfortunately generalizations of the most advanced methods such as perturbation theory (PT) and CC method turned out nontrivial. The problem is that in multi-reference cases we have to deal with two types of electron correlation, nondynamic and dynamic, that must be treated in different ways. The first type is associated with the strongly interacting reference determinants. The second one gives contributions to the wave function from the space orthogonal to that spanned by the reference functions. The wave function must include all reference determinants while the complementary space contributions should be incorporated in an approximate yet, as much as possible, effective way. Obviously, we would like to have them expressed through some kind of perturbation or coupled cluster expansion.

The multi-reference strategy can be easily implemented within the configuration interaction framework. The reason is that both the reference and outer space contributions appear in the wave function linearly so there is no difference in determining them. Thus the multireference generalization of CI is conceptually simple and straightforward. Some additional techniques like graphical unitary group approach (GUGA) have been developed [10] to make MR-CISD calculations simpler and more efficient. The only difference, compare to the SR case, is that now the eigenvalue problem of matrix representation of the Hamiltonian in a space spanned by reference determinants and determinants that are singly or doubly excited with respect to at least one them must be solved. Due to its simplicity MR-CISD is one of few methods that are used in routine calculations requiring a MR description. The disadvantages of MR-CISD are lack of size-extensivity and relatively poor description of dynamic The MR generalizations of many-body perturbation and CC theory are quite complicated and there are several reasons for that. First both types of correlation effects are treated differently in the expansion. Nondynamic correlation must be represented exactly in the wave function, i.e, through linear combination of the reference determinants while for the dynamic part truncated perturbation or coupled cluster expansions are assumed. It becomes clear that unlike MR-CISD this calls for a two-step procedures. There are two options that can be considered which differ in the order of determination of these effects. One can try to evaluate the nondynamic correlation first and then the impact of dynamic correlation on the energy. In such a case diagonalization of the Hamiltonian in the reference (model) space is performed first. Frequently the complete active space self consistent field (CAS SCF) method is used here. The orthogonal space contributions are assumed to be responsible for dynamic correlation and they are generated from the first order wave function obtained in the first step. These contributions are given by excitation (substitution) operators replacing occupied spin orbitals with unoccupied ones. Since some spin orbitals that are occupied in one reference determinant can be unoccupied in another one then such excitations can lead to linear dependencies. These linear dependencies must be then removed and that makes such an approach complicated and difficult to analyze. The advantage is that such schemes are quite similar to the SR ones. There are some examples of methods based on this idea like, contracted CI method [11] and a number perturbative approaches [12,13] with the CAS second-order perturbation theory (CASPT2) being probably most known and popular [13]. The CC schemes can be hardly formulated within this framework and only an attempt to derive and implement a linear version of MR-CC method of this kind has been made [14]. The reason is that cluster operators should be defined as second-quantized operators and for that Fermi vacuum or vacua must be defined. Such a definition introduces a unique classification of spin orbitals into occupied and unoccupied. The second option in formulating the two-step procedures is to evaluate the dynamic correlation first. To do this the effective Hamiltonian technique is used [15,16]. The so-called wave operator is introduced to give the exact wave function while acting on a linear combination of the reference determinants with The effective Hamiltonian acts within the reference space and its eigenvalues constitute a subset of eigenvalues of the Hamiltonian. The effective Hamiltonian technique allows us description of several states at a time. That makes the calculation numerically demanding and can be seen as a drawback of the formalism. Within the effective Hamiltonian framework there is a possibility to introduce Fermi vacuum to define excitation operators from reference determinants to the orthogonal complement of the reference space. Unfortunately unlike the SR case there is no obvious choice for the vacuum and basically two strategies can be followed.

The first one is to have one Fermi vacuum for excitations from all reference functions, the other is to use each reference determinant as a Fermi vacuum while defining excitations out of it. The two strategies lead to two formulations of the MR many-body perturbation theory (MR-MBPT). The first has been introduced by Brandow [17] while the second one by Hose and Kaldor [18].

As in the SR case the MBPT analysis of the wave operator can show an exponential structure of the expansion. Although the CC methods can be formulated without references to any kind of perturbation expansion they are frequently interpreted as approaches in which certain types of perturbative contributions to the wave function are summed to the infinite order. Thus with each of the two MR perturbation theory formulations, Brandow and Hose-Kaldor type, the corresponding CC expansions can be associated. The one that assume single Fermi vacuum to define cluster operators has been developed over many years with the most fundamental contributions given by Offermann, Ey, and Kümmel [19], Lindgren [20], and Mukherjee [21]. The most important concepts that have been introduced is the normal-ordered form of the exponential expansion [19,20] and valence universal strategy of solving the CC equations [21]. The method is known as valence-universal (VU) or Fockspace (FS) CC method. The second CC approach with cluster operators associated with different references has been proposed by Jeziorski and Monkhorst [22] and is called the state-universal (SU) or Hilbert-space (HS) CC method.

The other aspect that should be discussed here is the problem of size-extensivity of MR methods. The property is understood as a proper scaling of the energy with the number of particles. In the SR case size-extesivity can be associated with the connected diagram (term) theorem meaning that the energy is expressed through connected diagrams only.

In MR approaches the situation is more complicated because of the two-step procedure employed and both steps should analyzed as a potential source of size-inextenivity. A direct consequence of size-extensivity of a method is that the energy calculated for a supersystem consisting of two noninteracting subsystems is the sum of energies separately calculated for each of the subsystems. It has been found that diagonalization of the effective Hamiltonian can introduce size-extensivity error [23]. It was clear that as long as the reference space is complete diagonalization of the effective Hamiltonian should not hurt size-extensivity so such an assumption is usually made. More general reference spaces have been postulated in some cases but the one that has been quite widely applied is the quasi-complete space introduced by Mukherjee et al. [24] and Lindgren [25] for excitation energy calculations.

Finally problems that occur in practical applications should be mentioned. First of all selection of the reference space can be difficult in many cases. One of the requirements is that the reference space choice should lead to a clear separation of nondynamic and dynamic correlation effects otherwise the wave operator may have to deal with a significant contribution of nondynamic correlation and then problems with converging iterative procedures for solving the CC equations can occur. Jacobi-type perturbative schemes are especially vulnerable to improper selection of the reference space [26,27]. The other problems, the multi-reference CC methods are plagued with, are intruder states and multiple solutions [26,27,28,[START_REF] Paldus | Many-Body Methods in Quantum Chemistry[END_REF]30]. That means that can be some other state that interferes with the set of states we describe or that iterative schemes provide us with other solution of the CC equations than the desired one.

In the paper we would like to present recent attempts to obtain reliable and efficient methods capable of describing quasi-degenerate states. We start with the single reference approaches that have been modified to handle even a substantial degree of quasi-degeneracy.

. Extension of applicability of single-reference approaches to multi-reference cases

The method that is usually used to judge the significance of different contributions to wave function is the perturbation theory. The analysis shows that the most important part of electron correlation comes from double excitations. These can be easily supplemented with single excitations constituting the basic schemes with singles and doubles. Single and double excitations can be included linearly in the expansion or in the exponent which leads to the CISD and CCSD methods, respectively. When some degree of quasi-degeneracy is present then the perturbation expansion is not fast convergent and higher-order contributions must be included to maintain the same level of accuracy. These are, first of all, triple excitations but, since the computational cost of their full inclusion is high, their approximate influence on the energy through a posteriori corrections is evaluated within the CC framework. The inclusion of the lowest fourth-order contribution from triples leads to a scheme known as CCSD+T(CCSD) or CCSD[T] [6]. That can definitely improve the CCSD results when quasi-degeneracy is rather weak but the scheme breaks down completely when it becomes stronger. Inclusion of the fifth-order contribution do not help much so clearly SR perturbative arguments are not valid any more in such a situation and should be abandoned. As a result of many calculations it has been established that a reliable improvement is given by a combination of the CCSD[T] correction with one specific fifth-order term. That constitutes the CCSD(T) method [7] that has been broadly used in quantum chemical calculation allowing us, for example, quite reliable description of near equilibrium region in many molecular systems. It should be mentioned that some quadruple corrections have also been proposed like CCSDT(Q f ) and CCSD(TQ f ) [31,32].

The success of CCSD(T) has inspired some research activity towards extending applicability of a posteriori corrected truncated CC schemes even further. New schemes have been derived using the method of moments [33]. The method allows us to obtain a for- mula that gives a relation between the exact energy and the energy of a given truncated CC method provided the exact wave function is available. In practice the exact function is replaced by that obtained from the truncated CC calculation. A characteristic feature of the resulting corrections to the CC energies is the presence of renormalization denominators. Thus, depending on the scheme, corrections derived in this way are called the renormalized (R) or completely renormalized (CR) [34]. The basic types are known as R-CCSD(T) and CR-CCSD(T) [34]. It has been shown that such corrections can be easily found by allowing the energy to relax while deriving standard CCSD(T) correction so the method of moments is not really necessary to obtain them [35]. Currently the CR-CC(2,3) correction seems most reliable [36], however, we should be aware that cluster amplitudes that are used to calculate the corrections can be really large in quasi-degenerate situations thus terms that are neglected within some particular scheme are not negligible and, if included, can change the results significantly [37].

Methods mentioned above are of completely single-reference character meaning that they do not use a concept of the reference space. The CC methods that are basically singlereference but employ the idea of reference space are the state-specific MR-CC methods [38]. They can be described as the CCSDT or CCSDTQ methods in which, to reduce the size of the problem, only most important triple and quadruple excitations are included.

These are those which represent single or double excitations with respect to other reference determinants. Thus both the nodynamic and dynamic correlation are described through the cluster operator which can lead to large values of the cluster amplitudes. The recent abbreviations for the methods are CCSDt and CCSDtq which indicate that only selected triples and quadruples are included in the schemes.

The other approaches we would like to mention here are based on the MR-CISD method.

As mentioned in the introduction the MR-CI method retains formal and computational simplicity of its single-reference counterpart. Hence, the idea of using the MR-CISD scheme as a reliable source of information about nondynamic correlation and combine this with some SR ideas seems very appealing. As in the SR case one may try to introduce the effect of higher excitations than those included in MR-CISD through Davidson [START_REF] Davidson | The World of Quantum Chemistry[END_REF][START_REF] Pople | [END_REF]41]. Of course the MR-CISD method is much more difficult to analyze than the SR one so simple generalizations of the SR a posteriori corrections have been introduced without any deeper justification. Thus the Bruna-Peyerimhoff-Buenker correction can be viewed as an extension of the Davidson correction to the MR case [42,43] and the same has been done for two SR particle-number-dependent corrections [START_REF] Pople | [END_REF]41]. The other way of introducing the effect of higher excitations into the CI scheme that is known from the SR-CISD method is to modify the renormalization term to include only irreducible contributions i.e. those which are not canceled out when quadruple excitations are included. These methods are known as coupled electron pair approximations (CEPA) and several types of them have been developed [44]. The CEPA methods include only the so-called exclusion principle violation (EPV) terms and different types of CEPA include them to a different extent. This idea is difficult to apply within the MR-CISD scheme directly so it is convenient to introduce averaged evaluation of the EPV terms that has been done for the particle-number-dependent size-extensivity corrections to SR-CISD. Two corrections of this kind gave rise to two MR CEPA tapes methods: the MR average coupled pair functional (MR-ACPF) [45] and the MR averaged quadratic coupled cluster (MR-AQCC) [46]. They associate simple particle number dependent factors with the renormalization term in the MR-CISD equations. In spite of the fact that the factors are basically derived in the single reference context and thus can be considered as a very rough estimate of the effect of inclusion of higher excitations, both methods perform quite well in numerical applications.

The next category of methods we would like to concentrate on also combines MR-CISD with single-reference approaches. This type of methods have been initiated by Stolarczyk who proposed to introduce an information about nondynamic correlation into the SR-CC scheme from some simple but reliable external method [47]. He considered the CAS SCF method to be a good source of such an information. To be able to extract this information one of the determinants appearing in the CAS SCF function must be used as a single reference, all the others are treated as singly, doubly etc excited with respect to it. Such a reinterpretation of the CAS SCF function allows us to compare the method with other single reference methods like for example with the SR CC method. The linear CAS SCF expansion can now be equivalently expressed as an exponential one in the active space with the cluster amplitudes calculated from the CAS SCF coefficients. Simple relations between them allow to determine triple and quadruple cluster operators which can be used in the standard CC equations for singles and doubles. The so-called CC equations for singles and doubles can only contain up to quadruple cluster operators so the information about higher excitations than quadruple enters the equations through triples and quadruples. The singles and doubles depend on triples and quadruples while the energy on singles and doubles. This hierarchical structure allows to enter an external information to the CC equations. For example, if the exact triple and quadruple clusters are available then the exact energy can be obtained from the CC equations for singles and doubles. Having triples and quadruples extracted from the CAS SCF calculation one can transfer the information about nondynamic correlation obtained within CAS SCF into the CC scheme. Other external sources of this information, more or less reliable, can also be considered. Approaches of this type have been called the externally corrected CC methods [48]. Slightly different scheme has been proposed by Hino, Kinoshita and Bartlett under the name of tailored CC method [49] but the general idea underlying the approach is quite similar.

One of the most reliable yet still inexpensive sources of information on nondynamic correlation is MR-CISD. Disadvantages of MR-CISD are relatively poor description of the dynamic part of correlation and the lack of size-extensivity. An externally corrected CC scheme in which MR-CISD is used as a source of some triple and quadruple cluster amplitudes has been called the reduced multi-reference CC method (RMR-CCSD) [50]. It has been shown that the method is capable of providing good description of bond breaking situations where the degree of quasi-degeneracy is changing significantly with the geometry. It is well known that most of the genuine MR-CC approaches have a problem with a smooth transition from, for example, single-reference to multi-reference cases and that is one the main advantages of RMR-CCSD.

The RMR-CCSD method consists of several step. The first one is perform MR-CISD calculation for the ground state with properly selected reference space. The next is to take the Hartree-Fock determinant Φ as a reference and express the MR-CISD wave function in

Ψ MRCI = (1 + C 1 + C 2 + C ′ 3 + C ′ 4 + . . .)Φ, (1)
In addition to the single and double excitations, C 1 and C 2 , there are also some higher excitations present in the expansion. They represent single and double excitations with respect to other determinants from the reference space so only some of them appear in the MR-CISD wave function expressed in this way. This fact is indicated by adding prime to higher then double excitation operators in equation (1). From C clusters operators can be obtain using simple relations between CI coefficients and cluster amplitudes

C 1 = T 1 , (2)
C 2 = T 2 + (1/2)T 2 1 , (3)
C 3 = T 3 + T 1 T 2 + (1/3!)T 3 1 , (4)
C 4 = T 4 + T 1 T 3 + (1/2)T 2 2 + (1/2)T 2 1 T 2 + (1/4!)T 4 1 , (5)
and so on. The next step is to plug these amplitudes into the CC equations for singles and doubles

Φ 1 |[H exp(T 1 + T 2 + T ′ 3 )] c |Φ = 0, (6)
Φ 2 |[H exp(T 1 + T 2 + T ′ 3 + T ′ 4 )] c |Φ = 0, ( 7
)
where H is the Hamiltonian, Φ 1 and Φ 2 represent singly and doubly excited determinants, respectively, and [ ] c means connected terms only. For simplicity a nonlinear contribution to equation (7) involving [50]. Now an iterative procedure allows determination of T 1 and T 2 from the equations and calculation of the correlation energy from the standard CC energy expression.

T ′ 3 is neglected in RMR-CCSD
The RMR-CCSD scheme is a combination of MR-CISD and SR-CC in which the MR-CISD method is responsible for the nondynamic part of correlation while the SR-CC one takes care of reliable description of the dynamic part providing the reference space is properly selected in MR-CISD calculations. An alternative approach based on a similar strategy emerges when we assume that the MR-CISD method requires only some improvement of the dynamic correlation description and the reduction of size-extensivity error. A disadvantage of the MR-CISD scheme is that the linear expansion is used for both nondynamic and dynamic correlation so the description of dynamic part does not benefit from the efficiency of the exponential ansatz. The idea is to improve the MR-CISD wave function by adding additional terms that mimic the CC expansion. Since the correction concerns the dynamic part of correlation the SR CC expansion can be used . In the MR-CISD wave function, equation (1), only some higher than double excitation operators are included. Relying on the cluster analysis, equations ( 2)-( 3), which enable us determination of T 1 and T 2 from C 1 and C 2 , the remaining excitation operators can be constructed in the exponential fashion

Ψ M RCI+∆CC = (1 + C 1 + C 2 + C ′ 3 + C ′ 4 + . . . + Qe T 1 +T 2 )Φ, (8)
where

Q
E = E M RCI + Φ|C † 2 HQe T 1 +T 2 |Φ = E M RCI + ∆ D M RCI , (9)
if only the equation for C 2 is modified. The second term constitutes the MR-CISD energy correction. If the equation for C 1 is also changed according to equation ( 8) then we have [52,53]

∆ SD M RCI = ∆ D M RCI + Φ|C † 1 HQe T 1 +T 2 ]|Φ . ( 10
)
The corrections can be called the coupled cluster corrections (CCCs) to MR-CISD and can be seen as a noniterative version of the RMR-CCSD method. On the other hand they can be also considered as Davidson-type corrections. The corrections have been tested in many calculations and proved their effectiveness.

Their performance is very similar to that of RMR-CCSD method. They behave in a way similar to the regular CC methods meaning that they can handle quite significant degree of quasi-degeneracy but when the degree is too high they break down completely. The later can happen when the reference space in MR-CISD calculations is not sufficiently large to describe properly the nondynamic correlation thus the basic assumption is not fulfilled. being simultaneously moved apart and their distance from the initial position is used as a separation parameter. Calculation performed for 4H 2 includes FCI, CISD, CCSD, (2,2)MR-CISD and (2,2)MR-CISD+CCC. All the methods are exact for two-electron systems thus in the limit of full separation the energy given by a size-extensive method with respect FCI should be equal zero. Fig. 2 presents energies of the methods plotted against the separation parameter values. For small values of the separation parameter the ground state of 4H 2 is quasi-degenerate, the CCSD method does not perform well and there is a big difference between the CISD and (2,2)MR-CISD energies. The CC correction does a good job in reducing the MR-CISD energy error and giving a result very close to that of FCI. Then the degree of quasi-degeneracy rapidly decreases, the difference between CISD and MR-CISD becomes small and the CCSD method behaves like in typically single reference cases giving very good estimate of the exact energy. Already for the separation parameter value equal 3 a.u. the H 2 molecules can be considered well separated. It is seen that the only method that is fully size-extensive is CCSD reproducing the FCI energy. The CC corrected MR-CISD energy is not equal the FCI one but the difference given by MR-CISD is drastically reduced by the correction. Thus along with increasing the accuracy the CC correction is capable of reducing the inextensivity error of MR-CISD.

To summarize let us stress that in view of the fact that MR-CISD is one of few methods which are used in routine multi-reference calculations, the CC corrections, as a way of making MR-CISD more accurate and less size-inextensive seems very attractive. Finally it should be mentioned that some attempts have been made to use similar approaches in description of excited states [55, 56].

. Multi-reference MBPT and CC approaches

As mentioned in the introduction the so-called genuine MR approaches are based on the effective Hamiltonian formulation. We have found convenient and instructive to introduce the effective Hamiltonian formalism through simple similarity transformations [16]. Having the functional space partitioned into the reference space M 0 spanned by a set of reference determinants {Φ µ }, (µ = 1, . . . , m) and its orthogonal complement M ⊥ with the projection operators P 0 and Q, respectively, we can introduce a nilpotent operator X ≡ QXP 0 (X 2 = 0) and use it to perform the following similarity transformation of the Hamiltonian H = e -X He X = (1 -X)H(1 + X).

(11)

Requiring X to satisfy

Q HP 0 = Q(1 -X)H(1 + X)P 0 = 0, (12)
we have the complete eigenvalue problem of H divided into two subproblems since now all eigenvalues of H can be obtained by separate diagonalizations of two diagonal blocks of H, P 0 HP 0 and Q HQ. Assuming that we are interested in a subset of m(= dim M 0 ) eigenvalues of H only it is enough to solve the P 0 HP 0 eigenvalue problem. The first step is, however, to find X that satisfies equation ( 12).

The P 0 HP 0 operator is called the effective Hamiltonian

H ef f = P 0 HP 0 = P 0 H(1 + X)P 0 (13)
Action of H ef f is restricted to reference (model) space and its eigenvalues constitute a subset of the eigenvalues of H. Which eigenvalues are included in the subset depends on X.

Equation ( 12) is satisfied by many different X since is quadratic in X. Different X fulfilling equation (12) provide different subsets of the H eigenvalues, thus the problem is to find such X that leads to a set of energies we are interested in. If that is difficult to achieve then we can have what is called the intruder state problem [57] or multiple solution problem [START_REF] Paldus | Many-Body Methods in Quantum Chemistry[END_REF].

The two-step procedure introduced by the effective Hamiltonian formalism permits for adequate treatment of both types of correlation effects. The X operator should be responsible for dynamic correlation while diagonalization of H ef f should provide the nondynamical correlation contribution. To obtain good quality results it is not necessary to fully satisfy equation (12) but, for example, to a certain order of some perturbation expansion. Similarly a truncation of the M ⊥ space to that spanned by determinants that are singly or doubly excited with respect to at least one reference function Φ µ can be used to approximately satisfy equation (12). That would be equivalent to MR-CISD but this time formulated within the effective Hamiltonian framework that provides several eigenvalues at a time. Finally a 15 To apply many-body techniques it is essential to have Fermi vacuum defined. Then all operators entering equation ( 12) and ( 13) can be expressed as many-body operators and an explicit form of the equations can be obtained. As discussed in the introduction basically two strategies can be followed. Having

XP 0 = m µ=1 XP µ , P µ = |Φ µ Φ µ |, (14)
one may see attractive to use Φ µ as a Fermi vacuum to define excitation operators in XP µ and have different Fermi vacua for different components of X. Thus in the X equation corresponding to Φ µ that is obtained from equation ( 12)

QH(1 + X) -XP 0 H(1 + X)P µ = 0, (15)
the first term is similar to that in the SR case in each equation. The difference is that all XP µ must excite outside the reference space. The second term, however, is quite complicated since it couples equations for different Φ µ . Another characteristic feature of the approach is that XP µ of the same excitation-rank generate different subspaces of M ⊥ for different µ.

The second possibility is to use one Fermi vacuum Φ for all reference determinants.

Hence, first the reference functions must be obtained from the vacuum by a sequence of particle-hole creation operators D µ

Φ µ = D µ Φ, (16)
and then orthogonal space determinants are reached by applying X that now must consists of particle-hole annihilation operators to annihilate the reference state generated by D µ and a sequence of particle-hole creation operators to give the orthogonal space contribution. Here excitations in X of the different particle-rank can lead to the same excited function. There are many possibilities for making choice for Φ and whether D µ consists of particle only, hole only, or particle-hole creation operators depend on this choice and the reference space selection.

16 There are two kinds of perturbation expansion associated with the above choices for the Fermi vacuum. The first one is represented by MR perturbation expansion proposed be Hose and Kaldor [18]. The general rules for obtaining diagrammatic representation of the expansion have then been supplemented with an easier to follow order-by-order derivation [58]. The MR perturbation theory that is based on the single vacuum has been formulated

by Brandow [17]. Here the core determinant is selected to be the vacuum and reference functions are obtained by creating a number of particles on valence levels in all possible ways.

Reference spaces generated in this way are complete. Obviously, D µ contains only valence particle creation operators. This kind of perturbation expansion is especially convenient for description of atomic and nuclear systems. Another version of the perturbation expansion is obtained when the Hartree-Fock determinant for a closed shell neutral system is selected to be the vacuum and the reference determinants are generated by D µ through creation of k particles on unoccupied active orbital levels and l holes on occupied active ones. Such reference spaces are usually denoted by (k, l). With properly selected active unoccupied and occupied orbital levels a convenient tool for direct determination of excitation energies is obtained. Such a reference space is not complete. The concept has been developed by Hague and Mukherjee [24] and Lindgren [25] mainly in the context of MR CC applications.

Within the perturbation theory framework it has been possible to show a link between singlereference PT calculation of excitation energies and multi-reference PT one and justify some approximations and simplifications made there [59].

The main focus of the MR developments has been on the MR-CC methods. In this case the use of different vacua for different reference functions Φ i leads to different cluster operators for different Φ µ .

(

+ X)P 0 = m µ=1 e T µ P µ . ( 1
) 17
The CC ansatz has been proposed by Jeziorski and Monkhorst [22] and the method is known as the Hilbert-space or state-universal. As mentioned the simplicity of the expansion causes that the first term in equation ( 15) looks like that in the SR-CC equations but makes the second (renormalization) term that couples cluster operators defined with respect to different vacua complicated. The method has been implemented and used in some small size 17 calculations [60,[START_REF] Paldus | Many-Body Methods in Quantum Chemistry[END_REF]61,62]. Also an interesting variant of the method, the two-determinant CC approach has been proposed, by Balkova and Bartlett [63]. Several generalizations of the method to include incomplete model spaces have been considered with quite recent involving the so-called C-condition [64]. The HS-CC scheme has also state-specific versions like the Brillouin-Wigner MR-CC (B-W MRCC) [65] or state-specific MR-CC (SS-MRCC) [66]. However, the problem all HS-CC approaches face is that for different µ subspaces of M ⊥ generated T µ of the same excitation rank are different. The importance of this fact for the quality of the results has been emphasized by Berkovic and Kaldor [62].

The Fock-space or valence-universal CC method has been developed for many years by atomic and nuclear physicists and quantum chemists. The concept of the core determinant as a Fermi vacuum seems quite natural especially for atoms and nuclei. There are two major problems with excitation operators from the reference space defined with respect to a single vacuum. Defined as a second-quantized operators they can contain particle-hole annihilation operators so if they are used as cluster operators in the exponent then contractions between them are possible. The normal-ordered form of the exponential expansion thus can be very complicated. To avoid that the normal-ordered form of exponential expansion has been introduced by Offermann, Ey and Kümmel [19] and in a very elegant way by Lindgren [20] (1 + X)P 0 = {e S }P 0 ,

where { } means that operators are in the normal order so contractions between them are not allowed. The second important concept is valence universal strategy of solving the CC equations which has been introduced by Mukherjee [21]. This concept has been crucial to solve the problem of redundant cluster operators. The problem is that many different cluster operators lead to the same excited determinant while acting on the same reference function Φ µ so the number of equations given by the effective Hamiltonian scheme is not large enough to determine all of them. Valence universal strategy assumes that the additional equations are provided by problems with lover number of quasi-particles than the desired one and shows that the equations can be solved in a hierarchical way. For example, for the (k, 0) problem (with core and k valence particles) one has to solve the (0, 0) problem first, then

(1, 0), etc. up to the final (k, 0) problem. In each of them cluster amplitudes from lower sectors of the Fock-space appear as known parameters so only those corresponding to a given sector must be determined. It should be noticed, however, that the CC equations can also be solved without valence-universal strategy employed but through generating perturbative contributions of certain types to the wave operator given by equation ( 18) [26].

The use of (k, 0) reference spaces is convenient for atomic calculations. Already the four-electron Beryllium atom can be considered as a system having quasi-degenerate ground state. Unfortunately it has been found that with iterative procedures based on Jacobi-type algorithms it is not possible to converge the CC equations. A closer look at the problem reveals that this can be associated with the presence of the so-called intruder state which energy is close to those which we intend to obtained by diagonalization of the effective Hamiltonian. Thus for many years the VU-CC calculation for Be was seen as a classical example of the intruder state problem that enable to obtain solutions within MR-CC schemes. It appears, however, that the CC equations can be solved when more efficient iterative techniques are employed. With the use of the Newton-Raphson scheme the convergence has been finally reached showing that the method is capable of providing high quality results [28]. The Newton-Raphson iterative algorithm for solving sets on nonlinear equations is, however, numerically demanding and depends much on the starting vector. A different approach that can provide an efficient way of solving the FS-CC equations can be found when the fact that FS-CC expansion, equation (18), due to valence strategy of solving the FS-CC equations, is linear in the unknown cluster operators S (k,0) in the (k, 0), k > 0, sector of the Fock-space. That suggests that diagonalization techniques can also be used to solve the equations. A scheme that emerges from this idea can be considered as an implementation of standard iterative schemes for solving the CC equations are not required at all. To arrive at the intermediate Hamiltonian formulation we use, as before, similarity transformations of H [16]. The transformation leading to H ef f , equation (12), is split into a sequence of two similarity transformations by dividing X into

X = Y + Z, Z = P I XP 0 , Y = (Q -P I )XP 0 , (19)
where P I is the projection on the intermediate space that is defined as a subspace of the orthogonal space reached by cluster operators corresponding to a given sector of the Fock space. Let it be the (k, 0) sector. Then M (k,0) I is a subspace that is reached by excitations given by S (k,0) P (k,0) . It follows that then Y (k,0) contains only cluster operators from lower sectors than the (k, 0) one since due to valence universal strategy the expansion is linear in S (k,0) at the (k, 0) level. From equation ( 12) we have

P HP = P (1 -Z)(1 -Y )H(1 + Y )(1 + Z)P, (20)
where P = P 0 + P I projects on the reference and intermediate space. Having the standard FS-CC equations satisfied (P I HP 0 = 0) all eigenvalues of P HP can be obtained by separate digonalizations of two diagonal blocks (P 0 HP 0 and P I HP I ). The first one is the effective Hamiltonian so eigenvalues of H ef f are among them. It is easy to see that the eigenvalues of P HP are also given by

P HP = P (1 -Y )H(1 + Y )P, (21)
since both P HP and P HP are related through similarity transformation in P . Thus eigenvalues of H ef f are among those given by are obtained simultaneously. Each eigenvalue can be obtained separately if diagonalization allowing single-root calculation is used and in this sense the method can be considered state-specific. Effective diagonalization procurers for non-symmetric matrices [68]allow to overcome the convergence and intruder state problems. Also roots corresponding to alternative solutions of the effective Hamiltonian formulation can be accessed without any problem.

H I = P (1 -Y )H(1 + Y )P = P H(1 + Y )P, (22)
All this has been shown in the IH FS-CC calculation for Be [69] and other atomic systems [70].

Another problem that can be conveniently solved with the use of FS-CC method is the calculation of excitation energies. More general quasi-complete (k, l) reference spaces are needed in this case, first of all the (1, 1) one since most of low lying excited states are dominated by single excitations. For excitation energy calculation also other methods like the equation of motion CC (EOM CC) [71,72] or CC linear response theory (CC LRT) [73] can be effectively employed. These methods are not, however, fully size-extensive since they use linear expansion to describe excited states so only the ground state is parametrized by the exponential one. The staring point in all these approaches is the SR-CC calculation for the ground state where the ground state function is exp(T )Φ and Φ is the Hartree-Fock determinant. The equation can be written as HΦ = EΦ, H = e -T He T .

In other words we are looking for such a cluster operator T for which Φ is the right eigenfunction of the similarity transformed Hamiltonian H corresponding to the ground state energy.

Again similarity transformation is convenient to explain why excited states energies can be obtained as eigenvalues of H in the space spanned by excited determinants only. This fact was not completely clear and the procedure of neglecting Φ in the excited state expansion was considered as an approximation. Indeed diagonalization of the matrix representation of H in such a space does not allow determination of excited state wave functions directly and some additional simple calculation is necessary to obtain them but from the point of view of similarity transformations Φ does not have to be included as long as energies of excited states are considered.

The EOM CCSD method (or equivalently CCSD LRT) can be expressed in the framework of the fourth order of perturbation theory [74]. When two separated systems are considered then local excitations are not affected by them, so the method can be called size-intensive, but nonlocal charge transfer excitation are not properly described. Also a minimal dressing to the EOM-CC matrix (CEPA-type) has been proposed to eliminate disconnected contributions [75]. The scheme has been implemented quite recently and the proper behavior of charge transfer excitations shown [76].

The FS-CC method with the (k, l) reference space has been proposed by Haque and

Mukherjee [24]. The intruder state problem can hurt calculations of this kind but again the intermediate Hamiltonian technique can be used to ease the problem [77,78]. It must be emphasized, however, that more general (k, l) version of the FS-CC method differs from the (k, 0) version in one important aspect. The (k, l) sectors cannot be considered as different sectors of the Fock-space since some of them describe systems with the same number of particles like, for example, (0, 0), (1, 1), (2, 2), . . . or (0, 1), (1, 2), (2, 3), . . ., thus the same problem can be approached many times. For example, the (0, 1) calculation, along with principle ionization potentials, can provide shake-up ionization potentials as an alternative solutions which are a primary goal of the (1, 2) calculation. To avoid that a sequence of similarity transformations can be introduced leading to elimination of excitations to the lower sectors in the valence universal hierarchy [79]. The most important is, however, the (1,1) problem that is, as discussed, separated from the (0, 0) one by taking H so the deexcitation operators can be omitted in S (1,1) . The IH formulation can be obtained in a way similar to the previous case but now the hierarchy of calculations required is (0.0), (1, 0), (0, 1) and

(1, 1). En explicit form of the IH FS-CC equations on the CCSD level of approximation can be found in Ref. [78]. The IH formulation reviles several interesting features of the FS-CC method. The first is that at the (1, 0) and (0, 1) levels the eigenvalues of H ef f do not depend on the choice of active orbital levels but the cluster operators that are needed at the (1, 1) level do. The eigenvalues corresponding to some particular state are the same regardless whether they appear in the principle solution of the FS-CCSD method or in the alternative The IH FS-CC method has been implemented on the CCSD level [78,80] and then extended to include three-body clusters (CCSDT) [81]. Note that SDT, due to specific definition of excitation operators, means one-, two-and three-body contributions to S (k,l) at all levels under consideration. Numerous calculations have shown that the increase of accuracy depends on the truncation scheme for the cluster operator and on the choice of active orbital levels. An example of typical behavior is shown in Fig. 3 where several vertical excitation energies for the H 2 O molecule are presented. They are obtained with the IH-FSCC method at the SD and SDT level of approximation employing two different reference spaces. Here (m, n) means that m lowest-lying unoccupied orbitals and n highest-lying occupied ones are selected as active. It can be seen that the inclusion of three-body cluster operators is important to increase the accuracy when relatively small reference space is used in the calculation but it is not so important when larger reference spaces are used. Also larger reference spaces usually provide better evaluation of excitation energies as observed for H 2 O in Fig. 3. The quality of the IH FSCC results is similar or slightly better than that of EOM CC but the FS-CC method offers size-extensive description not only of the ground state but also of excited states. The necessary condition for size-extensivity is a proper behavior of excitation energies in the limit of full separation of a system into noninteracting subsystems. As mentioned for excitations that are localized on subsytems excitation energies given by both methods in the calculation for the supersystem are the same as those obtained separately for the subsystems. This property is frequently called size-intensivity. However, those excitations that are really affected by the separation are of the charge-transfer character. For them the excitation energy should be equal to the sum of ionization potential and electron affinity. Some other approaches that also try to employ diagonalization to solve the FS-CC equations must be mentioned here. These are the so-called eigenvalue-independent partitioning technique that is used to derive an alternative form of the FS-CC equations [84] and very similar if not identical idea called the canonical form of the Bloch equation [85]. The similarity transformed EOM-CC method that is based on quite different derivation also leads to a very similar working equations [86]. Quite different implementation of the intermediate Hamiltonian technique to FS-CC has been proposed by Kaldor group [87].

. Conclusion

In this paper we present some recent multi-reference MBPT and CC developments with the main focus on those initiated at the Quantum Theory Project (QTP) University of Florida in Gainesville. Two approaches to the problem of quasi-degenerate and open-shell systems that are, in our opinion, very promising and provide robust computational schemes are described in more detail. They have been discussed in a broader context in order to show the advantages they offer. On the special occasion of 50 Years of the Quantum Theory Project the author would like to thank for the hospitality he has always experienced during his several visit to the Rod Bartlett group at QTP, first as a post-doc (1988)(1989)(1990)) and then as a visiting professor (1992,1994,1995). The unique scientific atmosphere of QTP has been always very inspiring to him. Many thanks to Rod Bartlett for his optimism and deep believe that all problems we straggle with can be solved.

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 4 determined in the second step. The second step consists of construction of the effective Hamiltonian, which depends on the wave operator, and its diagonalization.

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  projects out of the MR-CISD space. Thus contributions that are not present in the MR-CCSD function are now represented in a CC manner, i.e. through products of lower excitation rank cluster operators. If this modification of the wave function is expected to introduce a relatively moderate change in the description of the state then a noniterative treatment is justified resulting in a CC corrections to the MR-CISD energy. Usually the energy expectation value expression is used to reach this point and first of all the correction evaluates changes of the energy caused by modification of the wave function introduced in the so-called equation for C 2 . In this way we obtain[51]

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  Let us discuss properties of CCC on one numerical example which is the simultaneous stretching of H-O bonds in H 2 O. The DZ model for which FCI results are available is used [54]. The model is frequently employed to test the performance of different methods in presence of various degrees of quasi-degeneracy. Fig. 1. shows energies of CISD, CCSD, (2,2)MR-CISD, (2,2)RMR-CCSD, and (2,2)MR-CISD+CCC (CCC=∆ D M RCI ) relative to the FCI energy. (2,2) means that we have CAS with two active orbitals and two active electrons as a reference space. The results show that for 1 R e we have typical SR situation thus the (2,2)MR-CISD result does not differ much from that of CISD. At the same time both RMR-CCSD and CCC are capable of reproducing the CCSD energy. The real advantage of RMR-CCSD and CCC over MR-CISD and CCSD is visible for 1.5 R e and 2 R e . Having good description of both types of correlation in approaches combining MR-CISD and SR-CC good accuracy results are obtained in quasi-degenerate situations. Especially for 2 R e the improvement is quite spectacular where ∼ 32 mH MR-CISD energy gap is reduced to ∼ 3 mH. The important feature of the methods is that it allows to proceed smoothly from single-reference to multireference situations. It is well known that the genuine MR-CC methods have difficulties in describing different molecular geometries using one reference space. Moreover, contrary to the MR-CC schemes, the approaches presented above are state-selective and are not plagued with the intruder state problem.The CI methods are not size-extensive and it would be interesting to know to what an extent the CCC correction can reduce the inextensivity error. That can be followed on calculation performed for a model being a slight modification of the well known H8 system[43]. The model consists of four H 2 molecules with the internuclear distance fix at 2 a.u. initially arranged in the regular octagonal geometry. Then all H 2 molecules are 13

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 1933 the intermediate Hamiltonian technique to the FS-CC method. The idea standing behind the intermediate Hamiltonian formalism is to obtained a part of X, that can cause divergence, by diagonalization. Thus diagonalization of a slightly larger matrix than that of H ef f is assumed. The matrix represents the intermediate Hamiltonian and it has all eigenvalues of H ef f among its eigenvalues. Because of specific features of FS-CC the intermediate Hamiltonian version of FS-CC is especially simple and both cluster amplitudes and energies can be obtained exclusively by diagonalization of the intermediate Hamiltonian. Thus the

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 2033 where H I stands for the intermediate Hamiltonian (IH). The intermediate Hamiltonian version of FS-CC method is completely equivalent to the effective Hamiltonian one but offers an efficient way of solving the CC equations. The IH FS-CC formulation allows us to replace the two step procedure in the standard FC-CC method with a one step procedure in which eigenvalues and eigenvectors (from which cluster amplitudes can be determined)

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 2133 to make a comparison with the FS-CC approach possible. The comparison shows presence of disconnected contributions to the EOM CC effective Hamiltonian

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 423332 Fig.4shows the behavior of two lowest charge transfer excitations during separation of

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 2433130 Figure 1: Energies of CISD, CCSD, (2,2)MR-CISD, (2,2)RMR-CCSD and (2,2)MR-CISD+CCC relative to FCI energy obtained for the DZ model of simultanious streaching of H-O bonds in H 2 O molecule[54]. The plot is based on results from Ref.[51].

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 231 Figure 2: Energies of CISD, CCSD, (2,2)MR-CISD and CC corrected (2,2)MR-CISD relative to FCI energy obtained for the 4H 2 model. The 4H 2 model describes separation of four H 2 molecules arranged initially into regular octogonal geometry into nonintracting H 2 molecules.The plot is based on results from Ref.[53].

1 A 2

 2 Exp.

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 3323343333 Figure 3: Vertical excitations energies for H 2 O obtained with FS-CCSD and FS-CCSDT using POLI [83] basis set. Results for two refernce spaces, (3,3) and (6,3), are depicted where (m,n) means m lowest unoccupied orbital levels and n highest occupied orbital levels being active. The chart is based on results from Ref. [82].

  coupled cluster expansion can be used that proved so effective in describing dynamic correlation effects in SR cases. One can see that for approximate schemes it is important to have X that do not have to deal with nondynamic correlation effects.

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  ones. That also concerns the(1,1) sector. Another interesting observation is that the IH matrix elements can contain disconnected contributions. Connectedness of terms entering the CC equations is usually used to prove size-extensivity of a specific CC scheme. It can be shown, however, that here the disconnected contributions to IH do not hurt size-extensivity but, quite the contrary, are necessary to maintain size-extensivity since they cancel out disconnected terms generated by diagonalization of the IH matrix[78].

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