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Additional verses from Psalm 65 (64) to the Requiem introit can be found in the Graduale Simplex (p. 401):

Qui audis oratiónem, * ad te omnis caro véniet propter iniquitátem.
Etsi prævaluérunt super nos impietátes nostræ, * tu propitiáberis eis.
Beátus quem elegísti et assumpsísti, * inhabitátbit in átriis tuis.
Replébimur bonis domus tæ, * sanctitáte templi tui.

The Graduale Triplex indicates additional psalm verses for the Introit Requiem - but these are for the OF only, both because their texts are taken from the Neo-Vulgate, and also because they are set to the introit psalm-tone VI, not to the special psalm-tone for use in the EF Introit Requiem alone.

As Willi Apel notes in his 1958 work Gregorian Chant, 'Finally it may be noticed that the Introit of the Mass for the Dead, Requiem æternam [{L.U.} 1807], employs a "sixth tone" all its own, with recitation on a throughout, and with special formulae for the mediant, the second intonation, and the termination.'

Annoyingly, the book Versus psalmorum et canticorum, which contains extra psalm verses for Introits for use at EF Masses, does not contain any extra verses for the Requiem Introit (though it does for the Communion Lux æterna).

I have searched without success for extra psalm verses for the Requiem Introit, with text and music appropriate to the EF, and in default of finding any I have been bold enough to come up with my own.

First, as the EF psalm verse is Psalm 64:2-3 (combining two into one), the obvious choices for additional verses are those next following in Psalm 64:4-6, which in the Vulgate run as follows:

[v. 4] Verba iniquórum praevaluérunt super nos, * et impietátibus nostris tu propitiáberis.
(The words of the wicked have prevailed over us: * and thou wilt pardon our transgressions.)

[v. 5a] Beátus quem elegísti et assumpsísti: * inhabitábit in átriis tuis.
(Blessed is he whom thou hast chosen and taken to thee: * he shall dwell in thy courts.)

[vv. 5b-6a] Replébimur in bonis domus tuæ; * sanctum est templum tuum, mirábile in æquitáte.
We shall be filled with the good things of thy house; * holy is thy temple, wonderful in justice.

[6b] Exáudi nos, Deus, salutáris noster, spes ómnium fínium terræ, et in mari longe.
Hear us, O God our saviour, * who art the hope of all the ends of the earth, and in the sea afar off.

It seems to me that of these, vv. 5a and 6b are the more appropriate, while v. 4 is less appropriate given its first phrase. I assume that in the absence of a set rule, it would be acceptable to choose two out of these four to sing as extra verses, should they be needed, for the Introit of Requiem Masses.

Second, as to the unique tone of the Requeim Introit psalm verse: utilising the convenient gabc notation, the first intonation is (fg) (gf) (gh), which is the same as the usual introit psalm-tone 6; but before the flex, it is [for accent](g) (h.), which appears unique; after the flex, it runs (f) (g) (h), which looks like the second intonation, or mediant cadence after the asterisk, for introit psalm-tone 6; before that mediant cadence is the unique formula [always for the last four syllables presumably](g) (ixi) (g) (h.); after the mediant cadence, it runs (f) (gh), the same as introit psalm-tone 6; but the ending is (f) (gh) [for accent](g) [if needed](f) (f.), which looks like that for the ordinary psalm-tone 6.

Given all this, I have used gabc notation and handy online resources to set the Requiem with extra psalm verses suitable to the EF, which I append.

I too was thinking about this introit recently. Could the unique psalmtone be more the result of correctly accenting Hebrew words, rather than a way to set this particular introit apart from the rest of the corpus?

See, for example, the Appendix of Part I of Grammer of Plainsong (1905) from Stanbrook Abbey. This gives examples of how to accommodate Hebrew words with accents on the last syllable - like Jerusalém or Sión. †

This line of reasoning has probably been pointed out before, but it would seem that much of the idiosyncrasy of the tone is just to do with how to correctly point a flex and mediation on those two Hebrew words. The ending is unique - granted. I would think it follows logically that any further verses should be sung to the regular Mode 6 introit tone, just with the unique ending - unless any more Hebrew words appear.

† Incidentally, the methods outlined in this appendix are what I have now adopted for setting English texts to the Gregorian tones.

Can someone try pointing this? Kudos to the one who did, but the assumpsísti in the PDF seems a bit clunky.

(In any event, there always seems to be less time than one imagines at the Requiem introit, with no incense and no Psalm Judica me.)

Is not this Hebrew word treatment similar to the treatment of English pointing/melodic formulae in Anglican books? Sarum Tonale, works of Healy Willan, etc

I have put together a collection of various instances of the verse "Te decet hymnus", mainly from medieval manuscripts, and also some 19th to 21st century publications.

I still have to look into this, but from a quick glance it seems to me that there have been some variances of this psalm tone in use throughout the ages.

It's surprising to see the euouae in the Requiem verse!

JonathanKK, that "sounds" about right to me! (For some reason though, I would think to go to fa on the first syllable of atriis, and then so on the second and la the third.)