The note, which was addressed to his parents and siblings as well as his wife's mother and sister, also explained: 'I am sorry for my brutality but I had no other way.'
Hawe also reportedly requested that he not be forgiven for his actions.
Details from the five-page letter had previously been given to members of the jury but were not read out at the inquest.
It was found at the scene of the murders on August 29, alongside another note that was taped on the back door of the house which read: 'Please do not come in.'
Hawe also explained in the note that it was easier for his sons to be killed than for them to be subjected to the trauma of his own suicide.
He even left instructions that he be cremated, with his ashes thrown out to sea, and he specifically requested he not be buried as a Catholic.
Hawe also spoke of his career worries, referring at one point in the note to how his students perceived him.
Professor Harry Kennedy, clinical director at the Central Mental Hospital, was asked by Coroner Dr Mary Flanagan to review Mr Hawe's suicide note and reports from his therapist and GP.
He said: 'The counselling notes from March to June last year indicate that Alan Hawe was troubled.'
Professor Kennedy told the hearing that he believed that at the time Mr Hawe carried out the murder-suicide he had progressed from long-term depression to a severe depressive episode with psychotic symptoms.
'When people act in the course of severe mental illness, such as very severe psychotic mental illness, their judgment is severely impaired,' Professor Kennedy said.
The inquest concluded last month, with Hawe's death being officially recorded as suicide. The three children and Clodagh were unlawfully killed by Hawe.
After it was closed, a solicitor for Mary Coll and Jacqueline Connolly, Clodagh's mother and sister, said Hawe targeted his wife and eldest son first for fear they may fight back.
On the steps of the courthouse, lawyer Liam Keane said the killings were premeditated and calculated.
Flanked by the grief-stricken Mrs Coll and her surviving daughter Jacqueline, he said: 'It is clear from the evidence presented at the inquest that Clodagh and her boys were killed in a sequence that ensured that the eldest and most likely to provide effective resistance were killed first, and they were executed in a manner that rendered them unable to cry out for help.'
The jury of six women and one man returned verdicts of unlawful killing of Mrs Hawe and her three boys and suicide in the death of Mr Hawe.
The inquest was told the vice-principal last visited the psychotherapist and his GP on June 21 2016.
Mr McConnell said Mr Hawe gave no indication that he would harm himself or others. Dr Paula McKevitt said Mr Hawe attended her surgery complaining about a sore toenail.
He also told her he had washed his feet in bleach.
The GP said he was a little stressed about work and had not been sleeping.
Professor Harry Kennedy, clinical director at the Central Mental Hospital, was asked by Coroner Dr Mary Flanagan to review Mr Hawe's suicide note and reports from his therapist and GP.
He said: 'The counselling notes from March to June last year indicate that Alan Hawe was troubled.'
Professor Kennedy told the hearing that he believed that at the time Mr Hawe carried out the murder-suicide he had progressed from long-term depression to a severe depressive episode with psychotic symptoms.
'When people act in the course of severe mental illness, such as very severe psychotic mental illness, their judgment is severely impaired,' Professor Kennedy said.
Referring to his review of the suicide note and the GP's and therapist's reports, Prof Kennedy said: 'Hindsight is always a very unfair advantage.'
WHERE TO GET HELP:
If you are worried about your or someone else's mental health, the best place to get help is your GP or local mental health provider. However, if you or someone else is in danger or endangering others, call police immediately on 111.
OR IF YOU NEED TO TALK TO SOMEONE ELSE:
• LIFELINE: 0800 543 354 (available 24/7)
• SUICIDE CRISIS HELPLINE: 0508 828 865 (0508 TAUTOKO) (available 24/7)
• YOUTHLINE: 0800 376 633
• NEED TO TALK? Free call or text 1737 (available 24/7)
• KIDSLINE: 0800 543 754 (available 24/7)
• WHATSUP: 0800 942 8787 (1pm to 11pm)
• DEPRESSION HELPLINE: 0800 111 757
